A125815 q-Bell numbers for q=5; eigensequence of A022169, which is the triangle of Gaussian binomial coefficients [n,k] for q=5.

1, 1, 2, 9, 103, 3276, 307867, 89520089, 83657942588, 258923776689771, 2717711483011792407, 98702105953049319472394, 12629828399521800714941435773, 5784963467206342855747483263957541, 9613516698678314330032600987632336641122

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..60

Formula

a(n) = Sum_{k=0..n-1} A022169(n-1,k) * a(k) for n>0, with a(0)=1.

a(n) = Sum_{k>=0} 5^k * A125810(n,k). - Alois P. Heinz, Feb 21 2025

Example

The recurrence: a(n) = Sum_{k=0..n-1} A022169(n-1,k) * a(k)
is illustrated by:
a(2) = 1*(1) + 6*(1) + 1*(2) = 9;
a(3) = 1*(1) + 31*(1) + 31*(2) + 1*(9) = 103;
a(4) = 1*(1) + 156*(1) + 806*(2) + 156*(9) + 1*(103) = 3276.
Triangle A022169 begins:
1;
1, 1;
1, 6, 1;
1, 31, 31, 1;
1, 156, 806, 156, 1;
1, 781, 20306, 20306, 781, 1;
1, 3906, 508431, 2558556, 508431, 3906, 1; ...

Maple

b:= proc(o, u, t) option remember;
     `if`(u+o=0, 1, `if`(t>0, b(u+o, 0$2), 0)+add(5^(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=5: */ {a(n)=subst(B_q(n), q, 5)}

Crossrefs

Cf. A022169, A125810, A125811, A125812, A125813, A125814.

Column k=5 of A381369.

Sequence in context: A098610 A226391 A348858 * A132494 A136172 A012986

Adjacent sequences: A125812 A125813 A125814 * A125816 A125817 A125818

Keyword

nonn

Author

Paul D. Hanna, Dec 10 2006

Status

approved