A127926 G.f.: 1-q = Sum_{k>=0} a(k)*q^k*Faq(k+1,q), where Faq(n,q) is the q-factorial of n.

1, -1, 1, -2, 4, -7, 11, -18, 35, -76, 166, -358, 775, -1686, 3638, -7716, 16108, -33349, 69022, -143605, 301179, -636932, 1355855, -2896168, 6186750, -13183426, 27988755, -59197443, 124824911, -262699256, 552438175, -1162010894, 2446434685, -5156873960

Offset: 0

Author

Paul D. Hanna, Feb 06 2007

Keywords

• sign

Examples

			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.
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..450

Crossrefs

First column of A179750. [From Mats Granvik, Jul 26 2010]

Cf. A129273.

Programs

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))}
for(n=0,25,print1(a(n),", "))

Formula

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