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A060058
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Triangle of numbers related to A000330 (sum of squares) and A000364 (Euler numbers).
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13
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1, 1, 1, 1, 5, 5, 1, 14, 61, 61, 1, 30, 331, 1385, 1385, 1, 55, 1211, 12284, 50521, 50521, 1, 91, 3486, 68060, 663061, 2702765, 2702765, 1, 140, 8526, 281210, 5162421, 49164554, 199360981, 199360981, 1, 204
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table;
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OFFSET
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0,5
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COMMENTS
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Row sums give A060059. Columns give A000012 (powers of 1), A000330 (sum of squares), A060060-2 for m=0,...,4. Main diagonal gives Euler numbers A000364. See triangle A060074.
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LINKS
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Table of n, a(n) for n=0..37.
W. Lang, First 9 rows.
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FORMULA
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a(n, m) = a(n-1, m)+((n+1-m)^2)*a(n, m-1), a(n, -1) := 0, a(0, 0)=1, a(n, m)=0 if n<m.
a(n, m)= ay(n-m+1, m) if n >= m >= 0, with the rectangular array ay(n, m) := sum((j^2)*ay(j+1, m-1), j=1..n), n >= 0, m >= 1; input: ay(n, 0)=1 (iterated sums of squares).
G.f. for m-th column: 1/(1-x) for m=0, (x^m)*sum(A060063(m, k)*x^k, k=0..m)/(1-x)^(3*m+1), m >= 1.
Recursion for g.f.s for m-th column: (1-x)*G(m, x)= x*G''(m-1, x)- G'(m-1, x) + G(m-1, x)/x, m>=2; G(1, x)=x*(1+x)/(1-x)^4; the apostrophe denotes differentiation w.r.t. x. G(0, x)=1/(1-x). (W. Lang, added Feb 13 2004.)
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EXAMPLE
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{1}; {1,1}; {1,5,5,}; {1,14,61,61}; ...
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CROSSREFS
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Sequence in context: A011094 A204005 A075298 * A092766 A060074 A011501
Adjacent sequences: A060055 A060056 A060057 * A060059 A060060 A060061
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KEYWORD
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nonn,easy,tabl
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AUTHOR
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Wolfdieter Lang, Mar 16 2001
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STATUS
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approved
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