A171568 Riordan array (f(x), x*f(x)) where f(x) is the g.f. of A064613.

1, 3, 1, 10, 6, 1, 37, 29, 9, 1, 150, 134, 57, 12, 1, 654, 622, 318, 94, 15, 1, 3012, 2948, 1686, 616, 140, 18, 1, 14445, 14317, 8781, 3693, 1055, 195, 21, 1, 71398, 71142, 45625, 21132, 7075, 1662, 259, 24, 1, 361114, 360602, 238170, 118042, 44303, 12345, 2464, 332, 27, 1

Offset: 0

Author

Philippe Deléham, Dec 11 2009

Keywords

• nonn
• tabl

Comments

Equal to A171515*B = B*A104259, B = A007318.

Examples

			Triangle T(n,k) begins
[0]     1;
[1]     3,     1;
[2]    10,     6,     1;
[3]    37,    29,     9,     1;
[4]   150,   134,    57,    12,    1;
[5]   654,   622,   318,    94,   15,    1;
[6]  3012,  2948,  1686,   616,  140,   18,   1;
[7] 14445, 14317,  8781,  3693, 1055,  195,  21,  1;
[8] 71398, 71142, 45625, 21132, 7075, 1662, 259, 24, 1;
.
Production array begins
  3, 1
  1, 3, 1
  1, 1, 3, 1
  1, 1, 1, 3, 1
  1, 1, 1, 1, 3, 1
  1, 1, 1, 1, 1, 3, 1
- _Philippe Deléham_, Mar 05 2013
		

Crossrefs

Sum_{k=0..n} T(n,k)*x^k = A033543(n), A064613(n), A005572(n), A005573(n) for x = -1, 0, 1, 2 respectively.

Cf. A007318, A052179, A171589, A171515, A104259.

Programs

Maple

T := proc(n,k) option remember;
if n < 0 or k < 0 then 0 elif n = k then 1 else
T(n-1, k-1) + 3*T(n-1,k) + add(T(n-1, k+1+i), i=0..n) fi end:
for n from 0 to 8 do seq(T(n,k), k = 0..n) od; # Peter Luschny, Oct 16 2022

Mathematica

T[n_, k_] := T[n, k] = If[n < 0 || k < 0, 0, If[n == k, 1, T[n-1, k-1] + 3*T[n-1, k] + Sum[T[n-1, k+1+i], {i, 0, n}]]];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 23 2024, after Peter Luschny *)

Formula

T(n, 0) - T(n, 1) = 2^n.

T(n, k) = T(n-1, k-1) + 3*T(n-1, k) + Sum_{i=0..n} T(n-1, k+1+i). - Philippe Deléham, Feb 23 2012

Extensions

Corrected and extended by Peter Luschny, Oct 16 2022