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%I #10 Oct 24 2020 03:13:55
%S 1,3,30,586,17430,696744,34892228,2095250576,146470011822,
%T 11669877667640,1043022527852272,103294254944725680,
%U 11223660850862809960,1327297414140637610776,169690627501555713200460,23320015259500560303564736,3428111061331035575475494598,536769111685159965192282250632,89187403511916331132476542213808
%N O.g.f. A(x) satisfies: [x^n] exp( -n*A(x) ) / (1 - n*x)^n = 0, for n > 0.
%C It is remarkable that this sequence should consist entirely of integers.
%C a(n) is even for n > 2, with a(2^k + 1) = 2 (mod 4) for k >= 1 (conjecture).
%H Paul D. Hanna, <a href="/A319940/b319940.txt">Table of n, a(n) for n = 1..300</a>
%F a(n) ~ c * d^n * n! / n^2, where d = 9.669628447... and c = 0.0559981... - _Vaclav Kotesovec_, Oct 24 2020
%e O.g.f.: A(x) = x + 3*x^2 + 30*x^3 + 586*x^4 + 17430*x^5 + 696744*x^6 + 34892228*x^7 + 2095250576*x^8 + 146470011822*x^9 + ...
%e ILLUSTRATION OF DEFINITION.
%e The table of coefficients of x^k/k! in exp(-n*A(x))/(1 - n*x)^n begins:
%e n=1: [1, 0, -5, -178, -13983, -2082676, -500286245, ...];
%e n=2: [1, 2, 0, -344, -30592, -4460832, -1052294144, ...];
%e n=3: [1, 6, 45, 0, -46323, -7614918, -1758528063, ...];
%e n=4: [1, 12, 184, 2960, 0, -10429504, -2724259328, ...];
%e n=5: [1, 20, 495, 14050, 391505, 0, -3527335025, ...];
%e n=6: [1, 30, 1080, 44712, 2022912, 86720544, 0, ...];
%e n=7: [1, 42, 2065, 115556, 7166733, 472602158, 28883187781, 0, ...]; ...
%e in which the coefficient of x^n in row n forms a diagonal of zeros.
%e RELATED SERIES.
%e exp(A(x)) = 1 + x + 7*x^2/2! + 199*x^3/3! + 14929*x^4/4! + 2175121*x^5/5! + 516079351*x^6/6! + 179777047927*x^7/7! + ...
%e exp(-A(x)) = 1 - x - 5*x^2/2! - 163*x^3/3! - 13271*x^4/4! - 2012761*x^5/5! - 487790189*x^6/6! - 172048095115*x^7/7! + ...
%o (PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A; A[m] = Vec( exp(-m*x*Ser(A))/(1-m*x +x^2*O(x^m))^m)[m+1]/m ); A[n]}
%o for(n=1, 30, print1(a(n), ", "))
%Y Cf. A319938, A319939.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Oct 11 2018
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