A129273 G.f.: 1-q = Sum_{k>=0} a(k)*q^k * Faq(k+1,q)^2, where Faq(n,q) is the q-factorial of n.

1, -1, 2, -7, 26, -95, 344, -1256, 4654, -17470, 66234, -253192, 974992, -3778966, 14729200, -57683066, 226806148, -894791874, 3540105138, -14039128725, 55786507642, -222047783006, 885073034920, -3532110787193, 14110281656038

Offset: 0

Author

Paul D. Hanna, Apr 07 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)^2 begin as follows:
k=0: 1;
k=1: .. -1, -2,-1;
k=2: ....... 2, 8, 16,.. 20,.. 16,... 8,.... 2;
k=3: ......... -7,-42, -133, -294, -497,. -672, ...;
k=4: ............. 26,. 208,. 884, 2652,. 6266, ...;
k=5: .................. -95, -950,-5035,-18810, ...;
k=6: ........................ 344, 4128, 26144, ...;
k=7: ............................ -1256,-17584, ...;
k=8: .................................... 4654, ...;
Sums cancel along column j for j>1, leaving 1-q.
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..420
• Eric Weisstein's World of Mathematics, q-Factorial.

Crossrefs

Cf. A127926.

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))))^2),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) }^2.