A304643 - OEIS

%I #3 May 15 2018 16:53:09

%S 1,1,21,2075,427745,150754575,80775206341,61079788584715,

%T 61918201760905701,81032697606275994779,132999148265782603510745,

%U 267549402517056738883934727,647439631215495429552890390761,1855591663455916911410267165824087,6216559993885861267930628826256971069,24072412148295906199113974687972130690707,106699538321376193436754733217464490904934733

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

%e 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 ILLUSTRATION OF DEFINITION.

%e The table of coefficients of x^k in (1+x)^(n^3)/A(x) begins:

%e n=0: [1, -1, -20, -2034, -423216, -149819400, -80452969380, ...];

%e n=1: [1, 0, -21, -2054, -425250, -150242616, -80602788780, ...];

%e n=2: [1, 7, 0, -2166, -440034, -153263214, -81663489960, ...];

%e n=3: [1, 26, 304, 0, -470529, -161955486, -84652727166, ...];

%e n=4: [1, 63, 1932, 36334, 0, -174849912, -90924716676, ...];

%e n=5: [1, 124, 7605, 305466, 8541159, 0, -98844355155, ...];

%e n=6: [1, 215, 22984, 1626786, 85217850, 3329937702, 0, ...];

%e n=7: [1, 342, 58290, 6599344, 557724906, 37306986588, 1944420120804, 0, ...]; ...

%e 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 RELATED SERIES.

%e 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 The logarithmic derivative of the g.f. A(x) begins

%e 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 (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 for(n=0, 20, print1(a(n), ", "))

%Y Cf. A304191, A304644, A304398.

%K nonn

%O 0,3

%A _Paul D. Hanna_, May 15 2018