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%I #19 Oct 19 2020 06:27:56
%S 1,1,3,26,417,9726,295000,10946172,478392123,24001955894,
%T 1357178076996,85294057678956,5893597893045486,443851259961124476,
%U 36172543480754645712,3171024571792211972824,297496306299698019850371,29738036578363255676373606,3155172706300699135457477884,354114794234668864071564974988,41914947879716810639378379595146
%N O.g.f. A(x) satisfies: 0 = [x^n] exp( n^2 * Integral 1/A(x) dx ) / A(x), for n > 0.
%C Note: 0 = [x^n] exp( n * Integral 1/F(x) dx ) / F(x) holds for n > 0 when F(x) = 1 + x.
%C It is remarkable that this sequence should consist entirely of integers.
%C For n > 0, a(n) is odd iff n = 2^k for k >= 0.
%H Paul D. Hanna, <a href="/A305144/b305144.txt">Table of n, a(n) for n = 0..520</a>
%F a(n) ~ c * d^n * (n-1)!, where d = 4 / (-LambertW(-2*exp(-2)) * (2 + LambertW(-2*exp(-2)))) and c = 0.0604992010464118... - _Vaclav Kotesovec_, Oct 19 2020
%e O.g.f.: A(x) = 1 + x + 3*x^2 + 26*x^3 + 417*x^4 + 9726*x^5 + 295000*x^6 + 10946172*x^7 + 478392123*x^8 + 24001955894*x^9 + 1357178076996*x^10 + ...
%e ILLUSTRATION OF DEFINITION.
%e The table of coefficients of x^k in exp(n^2*Integral 1/A(x) dx)/A(x) begins:
%e n=0: [1, -1, -2, -21, -364, -8830, -273972, -10313037, -455135384, ...];
%e n=1: [1, 0, -3, -24, -390, -9264, -284235, -10625424, -466720254, ...];
%e n=2: [1, 3, 0, -35, -495, -10773, -318192, -11635020, -503631630, ...];
%e n=3: [1, 8, 25, 0, -700, -14272, -388269, -13599240, -573208625, ...];
%e n=4: [1, 15, 102, 371, 0, -19746, -525980, -17134953, -691326666, ...];
%e n=5: [1, 24, 273, 1904, 8136, 0, -716177, -23528472, -891395739, ...];
%e n=6: [1, 35, 592, 6381, 47945, 238403, 0, -31651620, -1235181962, ...];
%e n=7: [1, 48, 1125, 17080, 187110, 1536336, 8774025, 0, -1646095140, ...];
%e n=8: [1, 63, 1950, 39435, 583620, 6681714, 60092844, 389166915, 0, ...]; ...
%e in which the main diagonal is all zeros after the initial term, illustrating that 0 = [x^n] exp(n^2 * Integral 1/A(x) dx)/A(x), for n > 0.
%e RELATED SERIES.
%e 1/A(x) = 1 - x - 2*x^2 - 21*x^3 - 364*x^4 - 8830*x^5 - 273972*x^6 - 10313037*x^7 - 455135384*x^8 - 22995056286*x^9 - 1307053358940*x^10 + ...
%e exp( Integral 1/A(x) dx) = 1 + x - x^3 - 6*x^4 - 78*x^5 - 1544*x^6 - 40605*x^7 - 1328178*x^8 - 51857806*x^9 - 2350025232*x^10 - 121120896906*x^11 - 6991877399100*x^12 + ..., which is an integer series.
%e A'(x)/A(x) = 1 + 5*x + 70*x^2 + 1557*x^3 + 46316*x^4 + 1705382*x^5 + 74365572*x^6 + 3732699789*x^7 + 211429236472*x^8 + 13318438851990*x^9 + 922595879008860*x^10 + ...
%o (PARI) {a(n) = my(A=[1], m); for(i=1, n+1, m=#A; A=concat(A, 0); A[m+1] = Vec( exp(m^2*intformal(1/Ser(A))) / Ser(A) )[m+1] ); A[n+1]}
%o for(n=0, 20, print1(a(n), ", "))
%Y Cf. A305137, A305138, A305139, A305140, A305141, A305142, A305143.
%Y Cf. A305145, A305146, A305147, A304861, A304862.
%K nonn
%O 0,3
%A _Paul D. Hanna_, May 31 2018
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