0,4

G.f.: 1-q = Sum_{k>=0} a(k)*q^k*Product_{i=1..k+1} (1-q^i)/(1-q).

Define Faq(n,q) = Product_{i=1..n} (1-q^i)/(1-q) for n>0, Faq(0,q)=1.

Then coefficients of q in a(k)*q^k*Faq(k+1,q) begin as follows:

k=0: 1;

k=1: .. -1, -1;

k=2: ....... 1, 2, 2,. 1;

k=3: ......... -2,-6,-10,-12,-10,. -6,. -2;

k=4: ............. 4, 16, 36, 60,. 80,. 88,.. 80, ...;

k=5: ................ -7,-35,-98,-203,-343, -497, ...;

k=6: .................... 11, 66, 220, 539, 1078, ...;

k=7: ....................... -18,-126,-486,-1368, ...;

k=8: ............................. 35, 280, 1225, ...;

k=9: ................................. -76, -684, ...;

k=10: ...................................... 166, ...;

Sums cancel down column j for j>1, leaving 1-q.

(PARI) {a(n)=if(n==0, 1, polcoeff(1-q- sum(k=0, n-1, a(k)*q^k*prod(j=1, k+1, (1-q^j)/(1-q+q*O(q^(n-k))))), n, q))}

Adjacent sequences: A127923 A127924 A127925 this_sequence A127927 A127928 A127929

Sequence in context: A018063 A000570 A023426 this_sequence A078513 A024622 A034337

sign

Paul D. Hanna (pauldhanna(AT)juno.com), Feb 06 2007