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A317347 O.g.f. A(x) satisfies: [x^n] exp( n^3*x - n*A(x) ) = 0 for n >= 1.

%I A317347 #7 Aug 06 2018 05:41:32
%S A317347 1,9,552,85842,24653700,11219022936,7393496092416,6649411839351120,
%T A317347 7822998961379912592,11662362974001268456560,
%U A317347 21487905123054927319268352,47958258768575173308988367040,127523196462392124262710980808384,398397752352904475778061859746030080,1445051361690004153927005867189533921280
%N A317347 O.g.f. A(x) satisfies: [x^n] exp( n^3*x - n*A(x) ) = 0  for n >= 1.
%C A317347 It is remarkable that this sequence should consist entirely of integers.
%H A317347 Paul D. Hanna, <a href="/A317347/b317347.txt">Table of n, a(n) for n = 1..200</a>
%F A317347 a(n) ~ sqrt(1-c) * 3^(3*n-1) * n^(2*n - 3/2) / (sqrt(2*Pi) * c^n * (3-c)^(2*n-1) * exp(2*n)), where c = -LambertW(-3*exp(-3)) = 0.1785606278779211065968086697... = -A226750. - _Vaclav Kotesovec_, Aug 06 2018
%e A317347 O.g.f.: A(x) = x + 9*x^2 + 552*x^3 + 85842*x^4 + 24653700*x^5 + 11219022936*x^6 + 7393496092416*x^7 + 6649411839351120*x^8 + ...
%e A317347 such that [x^n] exp( n^3*x - n*A(x) ) = 0  for n >= 1.
%e A317347 ILLUSTRATION OF DEFINITION.
%e A317347 The table of coefficients in  begins:
%e A317347 n=1: [1, 0, -18, -3312, -2059236, -2957847840, -8077030651800, ...];
%e A317347 n=2: [1, 6, 0, -7056, -4281984, -6040453824, -16367904244224, ...];
%e A317347 n=3: [1, 24, 522, 0, -6980580, -9667325376, -25560523291464, ...];
%e A317347 n=4: [1, 60, 3528, 189792, 0, -14146669440, -37025599219200, ...];
%e A317347 n=5: [1, 120, 14310, 1679040, 181358460, 0, -51097553724600, ...];
%e A317347 n=6: [1, 210, 43992, 9173088, 1887214464, 358972896960, 0, ...]; ...
%e A317347 in which the coefficient of x^n in row n forms a diagonal of zeros.
%e A317347 RELATED SERIES.
%e A317347 exp(A(x)) = 1 + x + 19*x^2/2! + 3367*x^3/3! + 2074537*x^4/4! + 2969379361*x^5/5! + 8096147776171*x^6/6! + 37321188279552199*x^7/7! + ...
%o A317347 (PARI) {a(n) = my(A=[1], m); for(i=1, n+1, m=#A; A=concat(A, 0); A[m+1] = Vec( exp(m^3*x +x*O(x^#A)) / Ser(A)^m )[m+1]/m ); polcoeff( log(Ser(A)), n)}
%o A317347 for(n=1, 20, print1(a(n), ", "))
%Y A317347 Cf. A317344, A317346.
%K A317347 nonn
%O A317347 1,2
%A A317347 _Paul D. Hanna_, Jul 29 2018