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A060058 Triangle of numbers related to A000330 (sum of squares) and A000364 (Euler numbers). 13
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

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.

LINKS

Table of n, a(n) for n=0..37.

W. Lang, First 9 rows.

FORMULA

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.)

EXAMPLE

{1}; {1,1}; {1,5,5,}; {1,14,61,61}; ...

MATHEMATICA

a[_, -1] = 0; a[0, 0] = 1; a[n_, m_] /; n < m = 0; a[n_, m_] := a[n, m] = a[n-1, m] + (n+1-m)^2*a[n, m-1]; Table[a[n, m], {n, 0, 8}, {m, 0, n}] // Flatten (* Jean-Fran├žois Alcover, Jul 09 2013 *)

CROSSREFS

Sequence in context: A011094 A204005 A075298 * A092766 A060074 A229160

Adjacent sequences:  A060055 A060056 A060057 * A060059 A060060 A060061

KEYWORD

nonn,easy,tabl

AUTHOR

Wolfdieter Lang, Mar 16 2001

STATUS

approved

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Last modified April 18 03:24 EDT 2014. Contains 240688 sequences.