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A060058
Triangle of numbers related to A000330 (sum of squares) and A000364 (Euler numbers).
16
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
OFFSET
0,5
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=1..n} (j^2)*ay(j+1, m-1), 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_{k=0..m} A060063(m, k)*x^k)/(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). - Wolfdieter Lang, Feb 13 2004.
EXAMPLE
Triangle T(n, k) starts:
[0] 1;
[1] 1, 1;
[2] 1, 5, 5;
[3] 1, 14, 61, 61;
[4] 1, 30, 331, 1385, 1385;
[5] 1, 55, 1211, 12284, 50521, 50521;
[6] 1, 91, 3486, 68060, 663061, 2702765, 2702765;
[7] 1, 140, 8526, 281210, 5162421, 49164554, 199360981, 199360981;
MAPLE
T := proc(n, k) option remember; if k = 0 then 1 else if k = n then T(n, k-1) else (n - k + 1)^2 * T(n, k - 1) + T(n - 1, k) fi fi end:
seq(print(seq(T(n, k), k=0..n)), n=0..7); # Peter Luschny, Sep 30 2023
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-François Alcover, Jul 09 2013 *)
CROSSREFS
Cf. A060059 (row sums), A000364 (main diagonal Euler numbers).
Columns: A000012 (powers of 1), A000330 (sum of squares), A060060-2 for m=0,...,4.
See triangle A060074.
Sequence in context: A204005 A075298 A370262 * A092766 A288389 A060074
KEYWORD
nonn,easy,tabl
AUTHOR
Wolfdieter Lang, Mar 16 2001
STATUS
approved