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A304643 G.f. A(x) satisfies: [x^n] (1+x)^(n^3) / A(x) = 0 for n>0.

%I A304643 #3 May 15 2018 16:53:09
%S A304643 1,1,21,2075,427745,150754575,80775206341,61079788584715,
%T A304643 61918201760905701,81032697606275994779,132999148265782603510745,
%U A304643 267549402517056738883934727,647439631215495429552890390761,1855591663455916911410267165824087,6216559993885861267930628826256971069,24072412148295906199113974687972130690707,106699538321376193436754733217464490904934733
%N A304643 G.f. A(x) satisfies: [x^n] (1+x)^(n^3) / A(x) = 0 for n>0.
%e A304643 G.f.: A(x) = 1 + x + 21*x^2 + 2075*x^3 + 427745*x^4 + 150754575*x^5 + 80775206341*x^6 + 61079788584715*x^7 + 61918201760905701*x^8 + ...
%e A304643 ILLUSTRATION OF DEFINITION.
%e A304643 The table of coefficients of x^k in (1+x)^(n^3)/A(x) begins:
%e A304643 n=0: [1, -1, -20, -2034, -423216, -149819400, -80452969380, ...];
%e A304643 n=1: [1, 0, -21, -2054, -425250, -150242616, -80602788780, ...];
%e A304643 n=2: [1, 7, 0, -2166, -440034, -153263214, -81663489960, ...];
%e A304643 n=3: [1, 26, 304, 0, -470529, -161955486, -84652727166, ...];
%e A304643 n=4: [1, 63, 1932, 36334, 0, -174849912, -90924716676, ...];
%e A304643 n=5: [1, 124, 7605, 305466, 8541159, 0, -98844355155, ...];
%e A304643 n=6: [1, 215, 22984, 1626786, 85217850, 3329937702, 0, ...];
%e A304643 n=7: [1, 342, 58290, 6599344, 557724906, 37306986588, 1944420120804, 0, ...]; ...
%e A304643 in which the main diagonal is all zeros after the initial term, illustrating that [x^n] (1+x)^(n^3) / A(x) = 0 for n>0.
%e A304643 RELATED SERIES.
%e A304643 1 - 1/A(x) = x + 20*x^2 + 2034*x^3 + 423216*x^4 + 149819400*x^5 + 80452969380*x^6 + 60910650903564*x^7 + 61792107766345152*x^8 + ...
%e A304643 The logarithmic derivative of the g.f. A(x) begins
%e A304643 A'(x)/A(x) = 1 + 41*x + 6163*x^2 + 1701881*x^3 + 751428751*x^4 + 483682989449*x^5 + 426965933360359*x^6 + 494840882952869729*x^7 + ...
%o A304643 (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-1)^3)/Ser(A) )[m] ); A[n+1]}
%o A304643 for(n=0, 20, print1(a(n), ", "))
%Y A304643 Cf. A304191, A304644, A304398.
%K A304643 nonn
%O A304643 0,3
%A A304643 _Paul D. Hanna_, May 15 2018