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%I #9 Oct 19 2020 06:27:18
%S 1,2,4,32,512,12000,366400,13688960,602193152,30397531136,
%T 1728411805184,109177081065472,7578667350118400,573143826340921344,
%U 46886796648225349632,4124437046595970498560,388153835886455237115904,38910750374376922179960832,4139100381105952654252048384,465644313330130076144183017472
%N O.g.f. A(x) satisfies: 0 = [x^n] exp( n*(n+1) * Integral 1/A(x) dx ) / A(x), for n > 0.
%C Note: 0 = [x^n] exp( n * Integral 1/G(x) dx ) / G(x) holds for n > 0 when G(x) = 1 + x.
%C Note: 0 = [x^n] exp( (n-1) * Integral 1/F(x) dx ) / F(x) holds for n > 0 when F(x) = sqrt(1 + x^2).
%C It is remarkable that this sequence should consist entirely of integers.
%H Vaclav Kotesovec, <a href="/A304862/b304862.txt">Table of n, a(n) for n = 0..340</a>
%F a(n) ~ c * d^n * (n-1)!, where d = 4 / (-LambertW(-2*exp(-2)) * (2 + LambertW(-2*exp(-2)))) and c = 0.08310334422... - _Vaclav Kotesovec_, Oct 19 2020
%e O.g.f.: A(x) = 1 + 2*x + 4*x^2 + 32*x^3 + 512*x^4 + 12000*x^5 + 366400*x^6 + 13688960*x^7 + 602193152*x^8 + 30397531136*x^9 + 1728411805184*x^10 + 109177081065472*x^11 + 7578667350118400*x^12 + 573143826340921344*x^13 + 46886796648225349632*x^14 + 4124437046595970498560*x^15 + ...
%e ILLUSTRATION OF DEFINITION.
%e The table of coefficients of x^k in exp(n*(n+1) * Integral 1/A(x) dx)/A(x) begins:
%e n=0: [1, -2, 0, -24, -400, -10080, -319872, -12251008, ...];
%e n=1: [1, 0, -4, -80/3, -456, -165536/15, -3089536/9, ...];
%e n=2: [1, 4, 0, -48, -616, -66816/5, -1985184/5, ...];
%e n=3: [1, 10, 36, 0, -976, -93312/5, -500928, ...];
%e n=4: [1, 18, 140, 1648/3, 0, -83680/3, -6379648/9, ...];
%e n=5: [1, 28, 360, 2736, 12200, 0, -1023072, ...];
%e n=6: [1, 40, 756, 8880, 70664, 1800288/5, 0, ...];
%e n=7: [1, 54, 1400, 69256/3, 269184, 34495552/15, 599302144/45, 0, ...]; ...
%e in which the main diagonal is all zeros after the initial term, illustrating that 0 = [x^n] exp( n*(n+1) * Integral 1/A(x) dx ) / A(x), for n > 0.
%e RELATED SERIES.
%e 1/A(x) = 1 - 2*x - 24*x^3 - 400*x^4 - 10080*x^5 - 319872*x^6 - 12251008*x^7 - 548218368*x^8 - 28018713600*x^9 - 1608234580480*x^10 + ...
%e exp(Integral 1/A(x) dx) = 1 + x - x^2/2! - 5*x^3/3! - 143*x^4/4! - 10279*x^5/5! - 1265009*x^6/6! - 238548701*x^7/7! - 63550271455*x^8/8! - 22650892439183*x^9/9! + ...
%e A'(x)/A(x) = 2 + 4*x + 80*x^2 + 1808*x^3 + 54912*x^4 + 2052736*x^5 + 90617984*x^6 + 4595611904*x^7 + 262620131840*x^8 + 16670924217344*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*(m+1)*intformal(1/Ser(A))) / Ser(A) )[m+1] );A[n+1]}
%o for(n=0,20,print1(a(n),", "))
%Y Cf. A304861, A305144, A305145, A305146, A305147.
%Y Cf. A305137, A305138, A305139, A305140, A305141, A305142, A305143.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jun 01 2018
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