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%I #12 Oct 19 2020 06:29:07
%S 1,1,7,97,1987,53281,1754245,68228209,3055471369,154724090845,
%T 8740256396563,545005932104377,37196779826275411,2759229671824346893,
%U 221140447146112986889,19051164839221523341825,1756309610450933072328241,172576908229287147075691417,18010455349270266144268806799,1989930676607696867000687913025
%N O.g.f. A(x) satisfies: 0 = [x^n] exp( n^2 * Integral A(x) dx ) / A(x), for n > 0.
%C It is remarkable that this sequence should consist entirely of integers.
%C 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 Paul D. Hanna, <a href="/A305137/b305137.txt">Table of n, a(n) for n = 0..300</a>
%F a(n) ~ c * d^n * (n-1)!, where d = 4 / (-LambertW(-2*exp(-2)) * (2 + LambertW(-2*exp(-2)))) and c = 0.2658290856... - _Vaclav Kotesovec_, Oct 19 2020
%e O.g.f.: A(x) = 1 + x + 7*x^2 + 97*x^3 + 1987*x^4 + 53281*x^5 + 1754245*x^6 + 68228209*x^7 + 3055471369*x^8 + 154724090845*x^9 + 8740256396563*x^10 + ...
%e ILLUSTRATION OF DEFINITION.
%e The table of coefficients of x^k in exp(n^2*Integral A(x) dx)/A(x) begins:
%e n=0: [1, -1, -6, -84, -1764, -48360, -1620186, -63857556, ...];
%e n=1: [1, 0, -6, -88, -1830, -249144/5, -1661842, -2284994352/35, ...];
%e n=2: [1, 3, 0, -90, -2025, -272391/5, -8967968/5, -488439972/7, ...];
%e n=3: [1, 8, 30, 0, -2100, -311856/5, -10175418/5, -545952984/7, ...];
%e n=4: [1, 15, 114, 532, 0, -327564/5, -2389194, -3186733572/35, ...];
%e n=5: [1, 24, 294, 2416, 13536, 0, -2539746, -763395912/7, ...];
%e n=6: [1, 35, 624, 7542, 68415, 2234001/5, 0, -4102900932/35, ...];
%e n=7: [1, 48, 1170, 19320, 242550, 12134424/5, 90334582/5, 0, ...]; ...
%e in which the main diagonal is all zeros after the initial term, illustrating that 0 = [x^n] exp(n^2*Integral A(x) dx)/A(x), for n > 0.
%e RELATED SERIES.
%e exp(Integral A(x) dx) = 1 + x + 2*x^2/2! + 18*x^3/3! + 648*x^4/4! + 50904*x^5/5! + 6700464*x^6/6! + 1310200848*x^7/7! + 354395417472*x^8/8! + 126396068810112*x^9/9! + ...
%e A'(x)/A(x) = 1 + 13*x + 271*x^2 + 7489*x^3 + 253771*x^4 + 10113877*x^5 + 461995381*x^6 + 23766009457*x^7 + 1359214691545*x^8 + 85572483605593*x^9 + ...
%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(Ser(A))) / Ser(A) )[m+1] );A[n+1]}
%o for(n=0,20,print1(a(n),", "))
%Y Cf. A305138, A305139, A305140, A305141, A305142, A305143.
%K nonn
%O 0,3
%A _Paul D. Hanna_, May 31 2018
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