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A305138 O.g.f. A(x) satisfies: 0 = [x^n] exp( n^3 * Integral A(x) dx ) / A(x), for n > 0.

%I A305138 #13 Oct 20 2020 02:07:53
%S A305138 1,1,29,2829,574365,198036917,103683001241,76732157303677,
%T A305138 76332092063105089,98308205372662506789,159202381329276508297845,
%U A305138 316697346580111339298407645,759241108161224318180212593541,2158996472269125350886176802590109,7185071088438157708730937024732618961,27665076438562174334456341670646413685101
%N A305138 O.g.f. A(x) satisfies: 0 = [x^n] exp( n^3 * Integral A(x) dx ) / A(x), for n > 0.
%C A305138 It is remarkable that this sequence should consist entirely of integers.
%C A305138 Note: 0 = [x^n] exp( n * Integral C(x) dx ) / C(x) holds for n > 0 when C(x) = 1 + x*C(x)^2 is a g.f. of the Catalan numbers (A000108).
%H A305138 Paul D. Hanna, <a href="/A305138/b305138.txt">Table of n, a(n) for n = 0..250</a>
%F A305138 a(n) ~ c * d^n * n!^2 / n^(3/2), where d = 27 / (-LambertW(-3*exp(-3)) * (3 + LambertW(-3*exp(-3)))^2) and c = 0.0614343742940117549375894089... - _Vaclav Kotesovec_, Oct 20 2020
%e A305138 O.g.f.: A(x) = 1 + x + 29*x^2 + 2829*x^3 + 574365*x^4 + 198036917*x^5 + 103683001241*x^6 + 76732157303677*x^7 + 76332092063105089*x^8 + ...
%e A305138 ILLUSTRATION OF DEFINITION.
%e A305138 The table of coefficients of x^k in exp(n^3*Integral A(x) dx)/A(x) begins:
%e A305138 n=0: [1, -1, -28, -2772, -567952, -196735000, -103247834508, ...];
%e A305138 n=1: [1, 0, -28, -8372/3, -570045, -2957867108/15, -930709619938/9, ...];
%e A305138 n=2: [1, 7, 0, -8512/3, -1754242/3, -1002113658/5, -4705750542584/45, ...];
%e A305138 n=3: [1, 26, 323, 0, -603988, -1046990252/5, -539032922442/5, ...];
%e A305138 n=4: [1, 63, 1988, 119140/3, 0, -3286250072/15, -5146650533948/45, ...];
%e A305138 n=5: [1, 124, 7722, 961184/3, 28298144/3, 0, -1087966178572/9, ...];
%e A305138 n=6: [1, 215, 23192, 1672776, 90300350, 18408322114/5, 0, ...]; ...
%e A305138 in which the main diagonal is all zeros after the initial term, illustrating that 0 = [x^n] exp(n^3*Integral A(x) dx)/A(x), for n > 0.
%e A305138 RELATED SERIES.
%e A305138 exp( Integral A(x) dx) = 1 + x + 2*x^2/2! + 62*x^3/3! + 17216*x^4/4! + 13870816*x^5/5! + 23847686176*x^6/6! + 74818727953312*x^7/7! + 387328675940041472*x^8/8! + ...
%e A305138 A'(x)/A(x) = 1 + 57*x + 8401*x^2 + 2284577*x^3 + 986920761*x^4 + 620790291801*x^5 + 536349430717661*x^6 + 610013974179245217*x^7 + ...
%o A305138 (PARI) {a(n) = my(A=[1],m); for(i=1,n+1, m=#A; A=concat(A,0); A[m+1] = Vec( exp(m^3*intformal(Ser(A))) / Ser(A) )[m+1] );A[n+1]}
%o A305138 for(n=0,20,print1(a(n),", "))
%Y A305138 Cf. A305137, A305139, A305140, A305141, A305142, A305143.
%K A305138 nonn
%O A305138 0,3
%A A305138 _Paul D. Hanna_, May 31 2018