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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.

%I A127926 #9 Mar 10 2023 09:02:55
%S A127926 1,-1,1,-2,4,-7,11,-18,35,-76,166,-358,775,-1686,3638,-7716,16108,
%T A127926 -33349,69022,-143605,301179,-636932,1355855,-2896168,6186750,
%U A127926 -13183426,27988755,-59197443,124824911,-262699256,552438175,-1162010894,2446434685,-5156873960
%N 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.
%H A127926 Paul D. Hanna, <a href="/A127926/b127926.txt">Table of n, a(n) for n = 0..450</a>
%F A127926 G.f.: 1-q = Sum_{k>=0} a(k)*q^k*Product_{i=1..k+1} (1-q^i)/(1-q).
%e A127926 Define Faq(n,q) = Product_{i=1..n} (1-q^i)/(1-q) for n>0, Faq(0,q)=1.
%e A127926 Then coefficients of q in a(k)*q^k*Faq(k+1,q) begin as follows:
%e A127926 k=0: 1;
%e A127926 k=1: .. -1, -1;
%e A127926 k=2: ....... 1, 2, 2,. 1;
%e A127926 k=3: ......... -2,-6,-10,-12,-10,. -6,. -2;
%e A127926 k=4: ............. 4, 16, 36, 60,. 80,. 88,.. 80, ...;
%e A127926 k=5: ................ -7,-35,-98,-203,-343, -497, ...;
%e A127926 k=6: .................... 11, 66, 220, 539, 1078, ...;
%e A127926 k=7: ....................... -18,-126,-486,-1368, ...;
%e A127926 k=8: ............................. 35, 280, 1225, ...;
%e A127926 k=9: ................................. -76, -684, ...;
%e A127926 k=10: ...................................... 166, ...;
%e A127926 Sums cancel down column j for j>1, leaving 1-q.
%o A127926 (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))}
%o A127926 for(n=0,25,print1(a(n),", "))
%Y A127926 First column of A179750. [From _Mats Granvik_, Jul 26 2010]
%Y A127926 Cf. A129273.
%K A127926 sign
%O A127926 0,4
%A A127926 _Paul D. Hanna_, Feb 06 2007