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%I #20 Oct 06 2020 03:38:59
%S 1,1,9,179,5661,249424,14337039,1035838044,91867414241,9833503227827,
%T 1253246430314670,187948018130914066,32818034910964227439,
%U 6608081830970361618546,1520982783352578794866344,397027611766464517915252056,116698001659938095895315068553,38375694701199964362412343063161
%N O.g.f. A(x) satisfies: [x^n] exp( n^2 * x*A(x) ) / A(x) = 0 for n > 0.
%C Note: [x^n] exp( n * x*G(x) ) / G(x) = 0 for n>0 when G(x) is the g.f. of A088716.
%C It is remarkable that this sequence should consist entirely of integers.
%C What is the limit A304402(n) / A304400(n) ? Seems to be near 1.51...
%C A304402(n) / A304400(n) tends to 1.522998920075488836991600223419379... - _Vaclav Kotesovec_, Oct 06 2020
%H Paul D. Hanna, <a href="/A304402/b304402.txt">Table of n, a(n) for n = 0..300</a>
%F a(n) ~ c * n!^2 * n^2, where c = 1.18365083976367345437640389636650727... - _Vaclav Kotesovec_, Oct 06 2020
%e O.g.f.: A(x) = 1 + x + 9*x^2 + 179*x^3 + 5661*x^4 + 249424*x^5 + 14337039*x^6 + 1035838044*x^7 + 91867414241*x^8 + 9833503227827*x^9 + ...
%e ILLUSTRATION OF DEFINITION.
%e The table of coefficients of x^k/k! in exp( n^2 * x*A(x) ) / A(x) begins:
%e n=0: [1, -1, -16, -972, -125952, -28275000, -9885939840, ...];
%e n=1: [1, 0, -15, -968, -125835, -28263864, -9883855835, ...];
%e n=2: [1, 3, 0, -860, -123456, -28073976, -9850185728, ...];
%e n=3: [1, 8, 65, 0, -104811, -26970576, -9680119083, ...];
%e n=4: [1, 15, 240, 3892, 0, -21937464, -9078485120, ...];
%e n=5: [1, 24, 609, 16528, 457173, 0, -7077136715, ...];
%e n=6: [1, 35, 1280, 49572, 2066880, 89033736, 0, ...];
%e n=7: [1, 48, 2385, 123880, 6839349, 411165624, 26124539077, 0, ...]; ...
%e in which the main diagonal is all zeros after the initial term, illustrating that [x^n] exp( n^2 * x*A(x) ) / A(x) = 0 for n > 0.
%e Terms along the secondary diagonal in the above table are divisible by the odd numbers: [1, 3/3, 65/5, 3892/7, 457173/9, 89033736/11, 26124539077/13, ...] = [1, 1, 13, 556, 50797, 8093976, 2009579929, ...].
%e RELATED SERIES.
%e exp( x*A(x) ) = 1 + x + 3*x^2/2! + 61*x^3/3! + 4537*x^4/4! + 702501*x^5/5! + 183891571*x^6/6! + 73567995313*x^7/7! + 42361186187601*x^8/8! + ...
%e The arithmetic inverse of the o.g.f. begins:
%e 1/A(x) = 1 - x - 8*x^2 - 162*x^3 - 5248*x^4 - 235625*x^5 - 13730472*x^6 - 1001798042*x^7 - 89479215104*x^8 - 9627430506669*x^9 + ...
%o (PARI) {a(n) = my(A=[1],m); for(i=1,n, A=concat(A,0); m=#A; A[m] = Vec( exp( (m-1)^2*x*Ser(A) ) / Ser(A) )[m] );A[n+1]}
%o for(n=0,20, print1(a(n),", "))
%Y Cf. A304400, A088716.
%K nonn
%O 0,3
%A _Paul D. Hanna_, May 25 2018
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