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 A304398 G.f. A(x) satisfies: [x^n] (1+x)^((n+1)^3) / A(x) = 0 for n>0. 2

%I

%S 1,8,199,19568,4309702,1628514128,927231430126,737350581437744,

%T 778840734924755140,1054020790695331268000,1778132840285207445942196,

%U 3659007006256230147804241040,9023119928096184018484024831288,26274442260784898029809836586675872,89218495222818281880277619804533375624,349496587851612327547463367678217875791792

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

%e 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 ILLUSTRATION OF DEFINITION.

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

%e n=0: [1, -7, -143, -17031, -4008021, -1560094653, -901603927833, ...;

%e n=1: [1, 0, -171, -18144, -4130451, -1588513680, -912609360075, ...;

%e n=2: [1, 19, 0, -20424, -4500552, -1670248944, -943515644316, ...;

%e n=3: [1, 56, 1369, 0, -5042565, -1848681000, -1008460310529, ...;

%e n=4: [1, 117, 6615, 221979, 0, -2071834128, -1129354648380, ...;

%e n=5: [1, 208, 21357, 1424544, 64174929, 0, -1267137137679, ...;

%e n=6: [1, 335, 55774, 6134466, 495645999, 29071716177, 0, ...; ...

%e 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 RELATED SERIES.

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

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

%Y Cf. A304193, A304399.

%K nonn

%O 0,2

%A _Paul D. Hanna_, May 14 2018

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Last modified December 3 02:46 EST 2021. Contains 349445 sequences. (Running on oeis4.)