Problem 57517. Easy Sequences 90: Triple Summation of a Combinatorial Function

Created by Ramon Villamangca

Solve Later

{"parts":[{"partUri":"/matlab/document.xml","contentType":"application/vnd.mathworks.matlab.code.document+xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\"?><w:document xmlns:w=\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\"><w:body><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>Given a positive integer </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>x</w:t></w:r></w:customXml><w:r><w:rPr><w:b/></w:rPr><w:t>, we are asked to evaluate the following triple summation:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>                </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/></w:customXmlPr><w:r><w:t>S(x) = \\sum_{y=0}^{x} \\sum_{n=0}^{y} \\sum_{k=0}^n \\frac_{(-4)^{k} {n \\choose k}}^{2k \\choose k}.</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>The symbol </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>n \\choose k</w:t></w:r></w:customXml><w:r><w:t> is the combination function, </w:t></w:r><w:r><w:rPr><w:b/><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>nchoosek</w:t></w:r><w:r><w:rPr/><w:t> in Matlab. Therefore, we can write the function in Matlab as follows:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[    >> S = @(x) sum(arrayfun(@(y) sum(arrayfun(@(n) sum(arrayfun(@(k) ((-4)^k*nchoosek(n,k)/nchoosek(2*k,k)),0:n)),0:y)),0:x));]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Hence, for </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>x=5</w:t></w:r></w:customXml><w:r><w:t>, we have:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[    >> S(5)\n    >> ans =\n       -1.3302]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>Please round-off the output to the nearest 4 decimal places.</w:t></w:r></w:p></w:body></w:document>","relationship":null}],"relationships":[{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/document","target":"/matlab/document.xml","relationshipId":"rId1"}]}

<div style = "text-align: start; line-height: 20.440001px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none; white-space: normal; "><div style="block-size: 313px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 563.71875px 156.5px; transform-origin: 563.71875px 156.5px; vertical-align: baseline; "><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="font-weight: 700; ">Given a positive integer </span></span><span style="font-family: STIXGeneral, STIXGeneral-webfont, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);">x</span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="font-weight: 700; ">, we are asked to evaluate the following triple summation:</span></span></div><div style="block-size: 57px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 28.5px; text-align: left; transform-origin: 384px 28.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">                </span></span><span style="vertical-align:-25px"><img 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" width="169" height="57" style="width: 169px; height: 57px;"></span></div><div style="block-size: 55px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 27.5px; text-align: left; transform-origin: 384px 27.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">The symbol </span></span><span style="vertical-align:-14px"><img src="data:image/png;base64,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" width="28.5" height="34" style="width: 28.5px; height: 34px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style=""> is the combination function, </span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="font-family: Menlo, Monaco, Consolas, &quot;Courier New&quot;, monospace; font-weight: 700; ">nchoosek</span></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style=""> in Matlab. Therefore, we can write the function in Matlab as follows:</span></span></div><div style="background-color: rgb(247, 247, 247); block-size: 20px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 560.71875px 10px; transform-origin: 560.71875px 10px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; "><div style="background-color: rgba(0, 0, 0, 0); border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; ">    &gt;&gt; S = @(x) sum(arrayfun(@(y) sum(arrayfun(@(n) sum(arrayfun(@(k) ((-4)^k*nchoosek(n,k)/nchoosek(2*k,k)),0:n)),0:y)),0:x));</span></span></div></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">Hence, for </span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" width="36.5" height="18" style="width: 36.5px; height: 18px;"></span><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="">, we have:</span></span></div><div style="background-color: rgb(247, 247, 247); block-size: 60px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 560.71875px 30px; transform-origin: 560.71875px 30px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; "><div style="background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 560.71875px 10px; transform-origin: 560.71875px 10px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; ">    &gt;&gt; S(5)</span></span></div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 560.71875px 10px; transform-origin: 560.71875px 10px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; ">    &gt;&gt; ans =</span></span></div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 1px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 1px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 1px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 1px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 560.71875px 10px; transform-origin: 560.71875px 10px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space: pre; margin-right: 45px; "><span style="margin-inline-end: 0px; margin-right: 0px; ">       -1.3302</span></span></div></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; "><span style="font-weight: 700; ">Please round-off the output to the nearest 4 decimal places.</span></span></div></div></div>

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Problem 57517. Easy Sequences 90: Triple Summation of a Combinatorial Function

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