Problem 1845. Pascal's pyramid

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{"relationships":[{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/document","targetMode":"","relationshipId":"rId1","target":"/matlab/document.xml"},{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/output","targetMode":"","relationshipId":"rId2","target":"/matlab/output.xml"}],"parts":[{"partUri":"/matlab/document.xml","relationship":[],"contentType":"application/vnd.mathworks.matlab.code.document+xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\"?>\n<w:document xmlns:w=\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\"><w:body><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>In Pascal's triangle each number is the sum of the two nearest numbers in the line above:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[ 1\n 1 1\n 1 2 1\n 1 3 3 1\n 1 4 6 4 1]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>A three-dimensional analog of Pascal's triangle can be defined as a square pyramid in which each number is the sum of the four nearest numbers in the layer above. Define a function</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>pascalp(n)</w:t></w:r><w:r><w:t> that returns the nth layer of this pyramid, as follows:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[ pascalp(1)\n 1\n pascalp(2)\n 1 1\n 1 1\n pascalp(3)\n 1 2 1\n 2 4 2\n 1 2 1\n pascalp(4)\n 1 3 3 1\n 3 9 9 3\n 3 9 9 3\n 1 3 3 1\n pascalp(5)\n 1 4 6 4 1\n 4 16 24 16 4\n 6 24 36 24 6\n 4 16 24 16 4\n 1 4 6 4 1]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Note: Pascal's pyramid can also be defined as a tetrahedron (see</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"http://en.wikipedia.org/wiki/Pascal%27s_pyramid\"><w:r><w:t>http://en.wikipedia.org/wiki/Pascal%27s_pyramid</w:t></w:r></w:hyperlink><w:r><w:t>), in which case the layers are triangular rather than square, and the numbers are the trinomial coefficients.</w:t></w:r></w:p></w:body></w:document>"},{"partUri":"/matlab/output.xml","contentType":"text/xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\" ?><embeddedOutputs><metaData><evaluationState>manual</evaluationState><layoutState>code</layoutState><outputStatus>ready</outputStatus></metaData><outputArray type=\"array\"/><regionArray type=\"array\"/></embeddedOutputs>"}]}

In Pascal's triangle each number is the sum of the two nearest numbers in the line above:<pre> 1
 1 1
 1 2 1
 1 3 3 1
 1 4 6 4 1</pre>A three-dimensional analog of Pascal's triangle can be defined as a square pyramid in which each number is the sum of the four nearest numbers in the layer above. 
Define a function <tt>pascalp(n)</tt> that returns the nth layer of this pyramid, as follows:<pre> pascalp(1)
 1
 pascalp(2)
 1 1
 1 1
 pascalp(3)
 1 2 1
 2 4 2
 1 2 1
 pascalp(4)
 1 3 3 1
 3 9 9 3
 3 9 9 3
 1 3 3 1
 pascalp(5)
 1 4 6 4 1
 4 16 24 16 4
 6 24 36 24 6
 4 16 24 16 4
 1 4 6 4 1</pre>Note: Pascal's pyramid can also be defined as a tetrahedron (see <a href = "http://en.wikipedia.org/wiki/Pascal%27s_pyramid">http://en.wikipedia.org/wiki/Pascal%27s_pyramid</a>), in which case the layers are triangular rather than square, and the numbers are the trinomial coefficients.

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Solution 309847

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Andrew Newell on 24 Aug 2013

Beautiful!

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Problem 1845. Pascal's pyramid

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