A125812 q-Bell numbers for q=2; eigensequence of A022166, which is the triangle of Gaussian binomial coefficients [n,k] for q=2.

1, 1, 2, 6, 28, 204, 2344, 43160, 1291952, 63647664, 5218320672, 719221578080, 168115994031040, 67159892835119296, 46166133463916209792, 54941957091151982047616, 113826217192695041078973184, 412563248965919999955196308224, 2627807814905396804499456018866688

Offset: 0

Author

Paul D. Hanna, Dec 10 2006

Keywords

• nonn

Examples

			The recurrence a(n) = Sum_{k=0..n-1} A022166(n-1,k) * a(k) is illustrated by:
a(2) = 1*(1) + 3*(1) + 1*(2) = 6;
a(3) = 1*(1) + 7*(1) + 7*(2) + 1*(6) = 28;
a(4) = 1*(1) + 15*(1) + 35*(2) + 15*(6) + 1*(28) = 204.
Triangle A022166 begins:
1;
1, 1;
1, 3, 1;
1, 7, 7, 1;
1, 15, 35, 15, 1;
1, 31, 155, 155, 31, 1;
1, 63, 651, 1395, 651, 63, 1; ...
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..90

Crossrefs

Cf. A022166, A125810, A125811, A125813, A125814, A125815.

Column k=2 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(2^(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

a[0] = 1; a[n_] := a[n] = Sum[QBinomial[n-1, k, 2] a[k], {k, 0, n-1}]; Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Apr 09 2016 *)

PARI

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

Formula

a(n) = Sum_{k=0..n-1} A022166(n-1,k) * a(k) for n>0, with a(0)=1.

a(n) = Sum_{k>=0} 2^k * A125810(n,k). - Alois P. Heinz, Feb 21 2025