A060058 Triangle of numbers related to A000330 (sum of squares) and A000364 (Euler numbers).

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

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=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

Adjacent sequences: A060055 A060056 A060057 * A060059 A060060 A060061

Keyword

nonn,easy,tabl

Author

Wolfdieter Lang, Mar 16 2001

Status

approved