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

0,2

Comments

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

Links

Table of n, a(n) for n=0..54.

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

Example

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

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 *)

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.

Sequence in context: A035324 A171814 A091965 * A107056 A365962 A337273

Adjacent sequences: A171565 A171566 A171567 * A171569 A171570 A171571

Keyword

nonn,tabl

Author

Philippe Deléham, Dec 11 2009

Extensions

Corrected and extended by Peter Luschny, Oct 16 2022

Status

approved