A110112 Square array of numbers associated to the recurrences b(k) = b(k-1) + n*b(k-2); array T(n,k), read by descending antidiagonals, for n, k >= 0.

1, 1, 1, 1, 3, 1, 1, 15, 5, 1, 1, 60, 55, 7, 1, 1, 260, 385, 133, 9, 1, 1, 1092, 3311, 1330, 261, 11, 1, 1, 4641, 25585, 18430, 3393, 451, 13, 1, 1, 19635, 208335, 210490, 68237, 7216, 715, 15, 1, 1, 83215, 1652145, 2673223, 1037673, 197456, 13585, 1065, 17, 1, 1

Offset: 0

Author

Paul Barry, Jul 12 2005

Keywords

• easy
• nonn
• tabl

Comments

Rows include A001655, (-1)^n*A015266(n+3), A110111.

Examples

			Array T(n,k) (with rows n >= 0 and columns k >= 0) begins as follows:
  1,  1,   1,    1,      1,       1,        1,          1, ...
  1,  3,  15,   60,    260,    1092,     4641,      19635, ...
  1,  5,  55,  385,   3311,   25585,   208335,    1652145, ...
  1,  7, 133, 1330,  18430,  210490,  2673223,   31940881, ...
  1,  9, 261, 3393,  68237, 1037673, 18598293,  300963537, ...
  1, 11, 451, 7216, 197456, 3761296, 89565861, 1842200151, ...
  ...
		

Crossrefs

Cf. A083856.

Programs

Maple

a := proc(n, k) local v; option remember; if k = 0 and 0 <= n then v := 0; end if; if k = 1 and 0 <= n then v := 1; end if; if 2 <= k and 0 <= n then v := a(n, k - 1) + n*a(n, k - 2); end if; v; end proc;
T := proc(n, k) a(n, k + 1)*a(n, k + 2)*a(n, k + 3)/(n + 1); end proc;
seq(seq(T(k,n-k), k=0..n), n=0..10); # Petros Hadjicostas, Dec 26 2019

Formula

T(n, k) = a(n, k+1) * a(n, k+2) * a(n, k+3)/(n+1), where a(n, k) is the solution to a(n, k) = a(n, k-1) + n*a(n, k-2) for k >= 2 with a(n, 0) = 0 and a(n, 1) = 1 for all n >= 0.

Row n has g.f. 1/((1 + n*x - n^3*x^2) * (1 - (3*n + 1)*x - n^3*x^2)).