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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.

%I A129273 #11 Feb 16 2025 08:33:05
%S A129273 1,-1,2,-7,26,-95,344,-1256,4654,-17470,66234,-253192,974992,-3778966,
%T A129273 14729200,-57683066,226806148,-894791874,3540105138,-14039128725,
%U A129273 55786507642,-222047783006,885073034920,-3532110787193,14110281656038
%N 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.
%H A129273 Paul D. Hanna, <a href="/A129273/b129273.txt">Table of n, a(n) for n = 0..420</a>
%H A129273 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/q-Factorial.html">q-Factorial</a>.
%F A129273 G.f.: 1-q = Sum_{k>=0} a(k)*q^k*{ Product_{i=1..k+1} (1-q^i)/(1-q) }^2.
%e A129273 Define Faq(n,q) = Product_{i=1..n} (1-q^i)/(1-q) for n>0, Faq(0,q)=1.
%e A129273 Then coefficients of q in a(k)*q^k * Faq(k+1,q)^2 begin as follows:
%e A129273 k=0: 1;
%e A129273 k=1: .. -1, -2,-1;
%e A129273 k=2: ....... 2, 8, 16,.. 20,.. 16,... 8,.... 2;
%e A129273 k=3: ......... -7,-42, -133, -294, -497,. -672, ...;
%e A129273 k=4: ............. 26,. 208,. 884, 2652,. 6266, ...;
%e A129273 k=5: .................. -95, -950,-5035,-18810, ...;
%e A129273 k=6: ........................ 344, 4128, 26144, ...;
%e A129273 k=7: ............................ -1256,-17584, ...;
%e A129273 k=8: .................................... 4654, ...;
%e A129273 Sums cancel along column j for j>1, leaving 1-q.
%o A129273 (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 A129273 for(n=0,25,print1(a(n),", "))
%Y A129273 Cf. A127926.
%K A129273 sign
%O A129273 0,3
%A A129273 _Paul D. Hanna_, Apr 07 2007