A127926 - OEIS

%I #9 Mar 10 2023 09:02:55

%S 1,-1,1,-2,4,-7,11,-18,35,-76,166,-358,775,-1686,3638,-7716,16108,

%T -33349,69022,-143605,301179,-636932,1355855,-2896168,6186750,

%U -13183426,27988755,-59197443,124824911,-262699256,552438175,-1162010894,2446434685,-5156873960

%N 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 Paul D. Hanna, <a href="/A127926/b127926.txt">Table of n, a(n) for n = 0..450</a>

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

%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) begin as follows:

%e k=0: 1;

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

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

%e k=3: ......... -2,-6,-10,-12,-10,. -6,. -2;

%e k=4: ............. 4, 16, 36, 60,. 80,. 88,.. 80, ...;

%e k=5: ................ -7,-35,-98,-203,-343, -497, ...;

%e k=6: .................... 11, 66, 220, 539, 1078, ...;

%e k=7: ....................... -18,-126,-486,-1368, ...;

%e k=8: ............................. 35, 280, 1225, ...;

%e k=9: ................................. -76, -684, ...;

%e k=10: ...................................... 166, ...;

%e Sums cancel down 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))))),n,q))}

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

%Y First column of A179750. [From _Mats Granvik_, Jul 26 2010]

%Y Cf. A129273.

%K sign

%O 0,4

%A _Paul D. Hanna_, Feb 06 2007