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

0,3

Links

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

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

Example

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; ...

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

Prog

(PARI) /* q-Binomial coefficients: */ {C_q(n, k)=if(n<k || k<0, 0, if(n==0 || k==0, 1, prod(j=n-k+1, n, 1-q^j)/prod(j=1, k, 1-q^j)))} /* q-Bell numbers = eigensequence of q-binomial triangle: */ {B_q(n)=if(n==0, 1, sum(k=0, n-1, B_q(k)*C_q(n-1, k)))} /* Eigensequence at q=4: */ {a(n)=subst(B_q(n), q, 4)}

Crossrefs

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

Column k=4 of A381369.

Sequence in context: A107270 A294351 A342177 * A295044 A374872 A305000

Adjacent sequences: A125811 A125812 A125813 * A125815 A125816 A125817

Keyword

nonn,changed

Author

Paul D. Hanna, Dec 10 2006

Status

approved