A125814 q-Bell numbers for q=4; eigensequence of A022168, which is the triangle of Gaussian binomial coefficients [n,k] for q=4.

1, 1, 2, 8, 72, 1552, 84416, 12107584, 4726583424, 5150624868864, 16010990175691264, 144648776120641766400, 3857411545088966609514496, 307705704204270334224705015808, 74294186209325019487040708053442560, 54874536782175258883045772243829235417088

Offset: 0

Author

Paul D. Hanna, Dec 10 2006

Keywords

• nonn

Examples

			The recurrence: a(n) = Sum_{k=0..n-1} A022168(n-1,k) * a(k) is illustrated by:
  a(2) = 1*(1) + 5*(1) + 1*(2) = 8;
  a(3) = 1*(1) + 21*(1) + 21*(2) + 1*(8) = 72;
  a(4) = 1*(1) + 85*(1) + 357*(2) + 85*(8) + 1*(72) = 1552.
Triangle A022168 begins:
  1;
  1,    1;
  1,    5,     1;
  1,   21,    21,      1;
  1,   85,   357,     85,     1;
  1,  341,  5797,   5797,   341,    1;
  1, 1365, 93093, 376805, 93093, 1365, 1;
  ...
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..65

Crossrefs

Cf. A022168, A125810, A125811, A125812, A125813, A125815.

Column k=4 of A381369.

Programs

Maple

b:= proc(o, u, t) option remember;
     `if`(u+o=0, 1, `if`(t>0, b(u+o, 0$2), 0)+add(4^(u+j-1)*
        b(o-j, u+j-1, min(2, t+1)), j=`if`(t=0, 1, 1..o)))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..18);  # Alois P. Heinz, Feb 21 2025

Mathematica

b[o_, u_, t_] := b[o, u, t] =
   If[u + o == 0, 1, If[t > 0, b[u + o, 0, 0], 0] + Sum[4^(u + j - 1)*
   b[o - j, u + j - 1, Min[2, t + 1]], {j, If[t == 0, {1}, Range[o]]}]];
a[n_] := b[n, 0, 0];
Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Mar 15 2025, after Alois P. Heinz *)

PARI

/* q-Binomial coefficients: */
{C_q(n,k)=if(n
				

Formula

a(n) = Sum_{k=0..n-1} A022168(n-1,k) * a(k) for n>0, with a(0)=1.

a(n) = Sum_{k>=0} 4^k * A125810(n,k). - Alois P. Heinz, Feb 21 2025