0,4

This sequence gives the number of terms in rows of A125810.

Row g.f.s B_q(n) of A125810 are polynomials in q generated by:

B_q(n) = Sum_{j=0..n-1} B_q(j) * C_q(n-1,j) for n>0 with B_q(0)=1

where the triangle of q-binomial coefficients C_q(n,k) begins:

1;

1, 1;

1, 1 + q, 1;

1, 1 + q + q^2, 1 + q + q^2, 1;

1, 1 + q + q^2 + q^3, 1 + q + 2*q^2 + q^3 + q^4, 1 + q + q^2 + q^3, 1;

The initial q-Bell coefficients in B_q(n) are:

B_q(0) = 1; B_q(1) = 1; B_q(2) = 2;

B_q(3) = 4 + q;

B_q(4) = 8 + 4*q + 3*q^2;

B_q(5) = 16 + 12*q + 13*q^2 + 8*q^3 + 3*q^4;

B_q(6) = 32 + 32*q + 42*q^2 + 38*q^3 + 33*q^4 + 15*q^5 + 10*q^6 + q^7.

Cq:= proc(n, k) local j; if n<k or k<0 then 0 elif n=0 or k=0 then 1 else mul(1-q^j, j=n-k+1..n)/mul(1-q^j, j=1..k) fi end: Bq:= proc(n) option remember; local k; if n=0 then 1 else simplify (add (Bq(k) * Cq(n-1, k), k=0..n-1)) fi end: a:= n-> nops (Bq(n)): seq (a(n), n=0..60); [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Aug 04 2009]

(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)))} /* Number of coefficients in B_q(n) as a polynomial in q: */ a(n)=#Vec(B_q(n))

Cf. A125810, A125812, A125813, A125814, A125815.

Sequence in context: A062485 A137179 A096777 this_sequence A071424 A008762 A101018

Adjacent sequences: A125808 A125809 A125810 this_sequence A125812 A125813 A125814

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Dec 10 2006

More terms from Alois P. Heinz (heinz(AT)hs-heilbronn.de), Aug 04 2009