A125813 - OEIS

%I #6 Jun 14 2017 00:18:43

%S 1,1,2,7,47,628,17327,1022983,132615812,38522717107,25526768401271,

%T 39190247441314450,141213238745969102393,1207367655155905204747681,

%U 24733467452839301566047854678,1224709126636123500201799360630423

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

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

%e The recurrence: a(n) = Sum_{k=0..n-1} A022167(n-1,k) * a(k)

%e is illustrated by:

%e a(2) = 1*(1) + 4*(1) + 1*(2) = 7;

%e a(3) = 1*(1) + 13*(1) + 13*(2) + 1*(7) = 47;

%e a(4) = 1*(1) + 40*(1) + 130*(2) + 40*(7) + 1*(47) = 628.

%e Triangle A022167 begins:

%e 1;

%e 1, 1;

%e 1, 4, 1;

%e 1, 13, 13, 1;

%e 1, 40, 130, 40, 1;

%e 1, 121, 1210, 1210, 121, 1;

%e 1, 364, 11011, 33880, 11011, 364, 1; ...

%o (PARI) /* q-Binomial coefficients: */

%o 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)))

%o /* q-Bell numbers = eigensequence of q-binomial triangle: */

%o B_q(n)=if(n==0,1,sum(k=0,n-1,B_q(k)*C_q(n-1,k)))

%o /* Eigensequence at q=3: */

%o a(n)=subst(B_q(n),q,3)

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

%K nonn

%O 0,3

%A _Paul D. Hanna_, Dec 10 2006