A129273 - OEIS

%I #10 Mar 10 2023 09:16:50

%S 1,-1,2,-7,26,-95,344,-1256,4654,-17470,66234,-253192,974992,-3778966,

%T 14729200,-57683066,226806148,-894791874,3540105138,-14039128725,

%U 55786507642,-222047783006,885073034920,-3532110787193,14110281656038

%N 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.

%H Paul D. Hanna, <a href="/A129273/b129273.txt">Table of n, a(n) for n = 0..420</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/q-Factorial.html">q-Factorial</a>.

%F G.f.: 1-q = Sum_{k>=0} a(k)*q^k*{ Product_{i=1..k+1} (1-q^i)/(1-q) }^2.

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

%e Then coefficients of q in a(k)*q^k * Faq(k+1,q)^2 begin as follows:

%e k=0: 1;

%e k=1: .. -1, -2,-1;

%e k=2: ....... 2, 8, 16,.. 20,.. 16,... 8,.... 2;

%e k=3: ......... -7,-42, -133, -294, -497,. -672, ...;

%e k=4: ............. 26,. 208,. 884, 2652,. 6266, ...;

%e k=5: .................. -95, -950,-5035,-18810, ...;

%e k=6: ........................ 344, 4128, 26144, ...;

%e k=7: ............................ -1256,-17584, ...;

%e k=8: .................................... 4654, ...;

%e Sums cancel along column j for j>1, leaving 1-q.

%o (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))}

%o for(n=0,25,print1(a(n),", "))

%Y Cf. A127926.

%K sign

%O 0,3

%A _Paul D. Hanna_, Apr 07 2007