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A304398 G.f. A(x) satisfies: [x^n] (1+x)^((n+1)^3) / A(x) = 0 for n>0.

%I A304398 #5 May 14 2018 12:37:14
%S A304398 1,8,199,19568,4309702,1628514128,927231430126,737350581437744,
%T A304398 778840734924755140,1054020790695331268000,1778132840285207445942196,
%U A304398 3659007006256230147804241040,9023119928096184018484024831288,26274442260784898029809836586675872,89218495222818281880277619804533375624,349496587851612327547463367678217875791792
%N A304398 G.f. A(x) satisfies: [x^n] (1+x)^((n+1)^3) / A(x) = 0 for n>0.
%e A304398 G.f.: A(x) = 1 + 8*x + 199*x^2 + 19568*x^3 + 4309702*x^4 + 1628514128*x^5 + 927231430126*x^6 + 737350581437744*x^7 + 778840734924755140*x^8 + ...
%e A304398 ILLUSTRATION OF DEFINITION.
%e A304398 The table of coefficients of x^k in (1+x)^((n+1)^3)/A(x) begins:
%e A304398 n=0: [1, -7, -143, -17031, -4008021, -1560094653, -901603927833, ...;
%e A304398 n=1: [1, 0, -171, -18144, -4130451, -1588513680, -912609360075, ...;
%e A304398 n=2: [1, 19, 0, -20424, -4500552, -1670248944, -943515644316, ...;
%e A304398 n=3: [1, 56, 1369, 0, -5042565, -1848681000, -1008460310529, ...;
%e A304398 n=4: [1, 117, 6615, 221979, 0, -2071834128, -1129354648380, ...;
%e A304398 n=5: [1, 208, 21357, 1424544, 64174929, 0, -1267137137679, ...;
%e A304398 n=6: [1, 335, 55774, 6134466, 495645999, 29071716177, 0, ...; ...
%e A304398 in which the main diagonal is all zeros after the initial term, illustrating that [x^n] (1+x)^((n+1)^3)/A(x) = 0 for n>0.
%e A304398 RELATED SERIES.
%e A304398 1 - 1/A(x) = 8*x + 135*x^2 + 16896*x^3 + 3991125*x^4 + 1556103528*x^5 + 900047824305*x^6 + 722051918333952*x^7 + 766786063398540525*x^8 + ...
%e A304398 The logarithmic derivative of the g.f. A(x) begins
%e A304398 A'(x)/A(x) = 8 + 334*x + 54440*x^2 + 16580278*x^3 + 7958081528*x^4 + 5480891617798*x^5 + 5107502440681208*x^6 + 6182250826385760238*x^7 + ...
%o A304398 (PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( (1+x +x*O(x^m))^(m^3)/Ser(A) )[m] ); A[n+1]}
%o A304398 for(n=0, 20, print1(a(n), ", "))
%Y A304398 Cf. A304193, A304399.
%K A304398 nonn
%O A304398 0,2
%A A304398 _Paul D. Hanna_, May 14 2018