OFFSET
0,3
FORMULA
a(n) = Sum_{k=0..n-1} A022169(n-1,k) * a(k) for n>0, with a(0)=1.
EXAMPLE
The recurrence: a(n) = Sum_{k=0..n-1} A022169(n-1,k) * a(k)
is illustrated by:
a(2) = 1*(1) + 6*(1) + 1*(2) = 9;
a(3) = 1*(1) + 31*(1) + 31*(2) + 1*(9) = 103;
a(4) = 1*(1) + 156*(1) + 806*(2) + 156*(9) + 1*(103) = 3276.
Triangle A022169 begins:
1;
1, 1;
1, 6, 1;
1, 31, 31, 1;
1, 156, 806, 156, 1;
1, 781, 20306, 20306, 781, 1;
1, 3906, 508431, 2558556, 508431, 3906, 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=5: */ {a(n)=subst(B_q(n), q, 5)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 10 2006
STATUS
approved