A125813 q-Bell numbers for q=3; eigensequence of A022167, which is the triangle of Gaussian binomial coefficients [n,k] for q=3.

1, 1, 2, 7, 47, 628, 17327, 1022983, 132615812, 38522717107, 25526768401271, 39190247441314450, 141213238745969102393, 1207367655155905204747681, 24733467452839301566047854678, 1224709126636123500201799360630423

Offset

0,3

Links

Table of n, a(n) for n=0..15.

Formula

a(n) = Sum_{k=0..n-1} A022167(n-1,k) * a(k) for n>0, with a(0)=1.

Example

The recurrence: a(n) = Sum_{k=0..n-1} A022167(n-1,k) * a(k)
is illustrated by:
a(2) = 1*(1) + 4*(1) + 1*(2) = 7;
a(3) = 1*(1) + 13*(1) + 13*(2) + 1*(7) = 47;
a(4) = 1*(1) + 40*(1) + 130*(2) + 40*(7) + 1*(47) = 628.
Triangle A022167 begins:
1;
1, 1;
1, 4, 1;
1, 13, 13, 1;
1, 40, 130, 40, 1;
1, 121, 1210, 1210, 121, 1;
1, 364, 11011, 33880, 11011, 364, 1; ...

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=3: */
a(n)=subst(B_q(n), q, 3)

Crossrefs

Cf. A022167, A125810, A125811, A125812, A125814, A125815.

Sequence in context: A117141 A349965 A305533 * A254439 A341214 A106159

Adjacent sequences: A125810 A125811 A125812 * A125814 A125815 A125816

Keyword

nonn

Author

Paul D. Hanna, Dec 10 2006

Status

approved