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, 58637855728567773833514895771659795097103477549

Offset: 0

Author

Paul D. Hanna, Dec 10 2006

Keywords

• nonn

Examples

			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;
  ...
		

Links

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

Crossrefs

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

Column k=5 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(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

Mathematica

b[o_, u_, t_] := b[o, u, t] =
   If[u + o == 0, 1, If[t > 0, b[u + o, 0, 0], 0] + Sum[5^(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} 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