OFFSET
0,3
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
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 10 2006
STATUS
approved