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, 18522, 948002, 28862471, 510964090, 4798037791, 19391512145, 19391512145

Offset: 0

Author

Wolfdieter Lang, Mar 16 2001

Keywords

• nonn
• easy
• tabl

Examples

			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;
  ...
		

Links

• Wolfdieter Lang, First 9 rows.

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.

Programs

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 *)

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