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A319943 O.g.f. A(x) satisfies: [x^n] exp( n^4*x - n^3*A(x) ) = 0 for n >= 1.

%I A319943 #9 Oct 13 2020 08:21:24
%S A319943 1,4,756,666368,1573194000,7732870168896,68447889079649744,
%T A319943 991668063561584680960,21964178018215273705692768,
%U A319943 706356702284500884771233120000,31676677475982382039405057763384448,1917371897548937365539316712575130025984,152498179997818599198940803864398810967251456
%N A319943 O.g.f. A(x) satisfies: [x^n] exp( n^4*x - n^3*A(x) ) = 0  for n >= 1.
%C A319943 It is remarkable that this sequence should consist entirely of integers.
%H A319943 Paul D. Hanna, <a href="/A319943/b319943.txt">Table of n, a(n) for n = 1..200</a>
%F A319943 a(n) ~ sqrt(1-c) * 2^(8*n - 7) * n^(3*n - 7/2) / (sqrt(Pi) * c^(n - 1/4) * (4-c)^(3*n - 3) * exp(3*n)), where c = -LambertW(-4*exp(-4)) = 0.079309605127113656439108647386463779474372628792... - _Vaclav Kotesovec_, Oct 13 2020
%e A319943 G.f.: A(x) = x + 4*x^2 + 756*x^3 + 666368*x^4 + 1573194000*x^5 + 7732870168896*x^6 + 68447889079649744*x^7 + 991668063561584680960*x^8 + ...
%e A319943 ILLUSTRATION OF DEFINITION.
%e A319943 The table of coefficients of x^k/k! in exp( n^4*x - n^3*A(x) ) begins:
%e A319943 n=1: [1, 0, -8, -4536, -15992640, -188782917120, ...];
%e A319943 n=2: [1, 8, 0, -37312, -129112064, -1515383749632, ...];
%e A319943 n=3: [1, 54, 2700, 0, -453396528, -5216886215136, ...];
%e A319943 n=4: [1, 192, 36352, 6492672, 0, -12944825376768, ...];
%e A319943 n=5: [1, 500, 249000, 122933000, 57869896000, 0, ...];
%e A319943 n=6: [1, 1080, 1164672, 1253133504, 1340717598720, 1376766221921280, 0, ...]; ...
%e A319943 in which the coefficient of x^n in row n forms a diagonal of zeros.
%e A319943 RELATED SERIES.
%e A319943 exp(A(x)) = 1 + x + 9*x^2/2! + 4561*x^3/3! + 16011217*x^4/4! + 188863653441*x^5/5! + 5568801588349081*x^6/6! + 345016372861620194449*x^7/7! + ...
%o A319943 (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^3) )[m+1]/m^3 ); polcoeff( log(Ser(A)), n)}
%o A319943 for(n=1, 15, print1(a(n), ", "))
%Y A319943 Cf. A319941, A319942, A319944, A317344, A317346, A317347.
%K A319943 nonn
%O A319943 1,2
%A A319943 _Paul D. Hanna_, Oct 02 2018