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

Offset

0,3

Links

Table of n, a(n) for n=0..16.

Formula

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

Example

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; ...

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 *)

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=2: */ {a(n)=subst(B_q(n), q, 2)}

Crossrefs

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

Sequence in context: A178446 A324126 A272662 * A093657 A355064 A305627

Adjacent sequences: A125809 A125810 A125811 * A125813 A125814 A125815

Keyword

nonn

Author

Paul D. Hanna, Dec 10 2006

Status

approved