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A319942 O.g.f. A(x) satisfies: [x^n] exp( n^4*x - n^2*A(x) ) = 0 for n >= 1.

%I A319942 #8 Oct 13 2020 08:17:30
%S A319942 1,18,5616,6776352,20200266000,119799079486272,1242032508354758400,
%T A319942 20634722510624457007104,515736410631216295520236032,
%U A319942 18480208067078637967802351884800,913939154183946975187574927409795072,60487962958244860971401604975128195088384
%N A319942 O.g.f. A(x) satisfies: [x^n] exp( n^4*x - n^2*A(x) ) = 0  for n >= 1.
%C A319942 It is remarkable that this sequence should consist entirely of integers.
%H A319942 Paul D. Hanna, <a href="/A319942/b319942.txt">Table of n, a(n) for n = 1..200</a>
%F A319942 a(n) ~ sqrt(1-c) * 2^(8*n - 9/2) * n^(3*n - 5/2) / (sqrt(Pi) * c^n * (4-c)^(3*n - 2) * exp(3*n)), where c = -LambertW(-4*exp(-4)) = 0.07930960512711365643910864738... - _Vaclav Kotesovec_, Oct 13 2020
%e A319942 G.f.: A(x) = x + 18*x^2 + 5616*x^3 + 6776352*x^4 + 20200266000*x^5 + 119799079486272*x^6 + 1242032508354758400*x^7 + ...
%e A319942 ILLUSTRATION OF DEFINITION.
%e A319942 The table of coefficients of x^k/k! in exp( n^4*x - n^2*A(x) ) begins:
%e A319942 n=1: [1, 0, -36, -33696, -162628560, -2424019789440, ...];
%e A319942 n=2: [1, 12, 0, -138240, -657040896, -9735157974528, ...];
%e A319942 n=3: [1, 72, 4860, 0, -1533920976, -22357116073728, ...];
%e A319942 n=4: [1, 240, 57024, 12870144, 0, -41496660080640, ...];
%e A319942 n=5: [1, 600, 359100, 213537600, 121570858800, 0, ...];
%e A319942 n=6: [1, 1260, 1586304, 1994264064, 2496165050880, 3006510865205760, 0, ...]; ...
%e A319942 in which the coefficient of x^n in row n forms a diagonal of zeros.
%e A319942 RELATED SERIES.
%e A319942 exp(A(x)) = 1 + x + 37*x^2/2! + 33805*x^3/3! + 162771337*x^4/4! + 2424857569561*x^5/5! + 86269983111064621*x^6/6! + 6260449705448367386917*x^7/7! + ...
%o A319942 (PARI) {a(n) = my(A=[1], m); for(i=1, n+1, m=#A; A=concat(A, 0); A[m+1] = Vec( exp(m^4*x +x*O(x^#A)) / Ser(A)^(m^2) )[m+1]/m^2 ); polcoeff( log(Ser(A)), n)}
%o A319942 for(n=1, 15, print1(a(n), ", "))
%Y A319942 Cf. A319941, A319943, A319944, A317344, A317345, A317346, A317347.
%K A319942 nonn
%O A319942 1,2
%A A319942 _Paul D. Hanna_, Oct 02 2018