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A305139 O.g.f. A(x) satisfies: 0 = [x^n] exp( n^2 * Integral A(x)^2 dx ) / A(x), for n > 0.

%I A305139 #11 Oct 20 2020 01:58:36
%S A305139 1,1,9,155,3805,118632,4429279,191275884,9340355265,507681357635,
%T A305139 30360217294454,1979895257720082,139811654124752231,
%U A305139 10629630950986800850,865954337592580081080,75286721276048241037848,6961094538227014053702537,682423909436661488354778945,70743106543492192940195723155,7736186700358670253328879658965
%N A305139 O.g.f. A(x) satisfies: 0 = [x^n] exp( n^2 * Integral A(x)^2 dx ) / A(x), for n > 0.
%C A305139 It is remarkable that this sequence should consist entirely of integers.
%C A305139 Note: 0 = [x^n] exp( n * Integral F(x)^2 dx ) / F(x) holds for n > 0 when F(x) = 1 + x*F(x)^3 is a g.f. of A001764.
%H A305139 Paul D. Hanna, <a href="/A305139/b305139.txt">Table of n, a(n) for n = 0..300</a>
%F A305139 a(n) ~ c * d^n * (n-1)!, where d = 4 / (-LambertW(-2*exp(-2)) * (2 + LambertW(-2*exp(-2)))) and c = 0.652748826558... - _Vaclav Kotesovec_, Oct 19 2020
%e A305139 O.g.f.: A(x) = 1 + x + 9*x^2 + 155*x^3 + 3805*x^4 + 118632*x^5 + 4429279*x^6 + 191275884*x^7 + 9340355265*x^8 + 507681357635*x^9 + ...
%e A305139 ILLUSTRATION OF DEFINITION.
%e A305139 The table of coefficients of x^k in exp(n^2*Integral A(x)^2 dx)/A(x) begins:
%e A305139 n=0: [1, -1, -8, -138, -3440, -108905, -4118952, -179740162, ...];
%e A305139 n=1: [1, 0, -15/2, -140, -28065/8, -111009, -67094895/16, -1279321635/7, ...];
%e A305139 n=2: [1, 3, 0, -130, -3660, -117117, -4419200, -1344147030/7, ...];
%e A305139 n=3: [1, 8, 65/2, 0, -27265/8, -124886, -76650687/16, -2911952885/14, ...];
%e A305139 n=4: [1, 15, 120, 630, 0, -117081, -5202600, -1614205230/7, ...];
%e A305139 n=5: [1, 24, 609/2, 2712, 139455/8, 0, -78693087/16, -3562210803/14, ...];
%e A305139 n=6: [1, 35, 640, 8190, 81940, 620323, 0, -1698895510/7, ...];
%e A305139 n=7: [1, 48, 2385/2, 20540, 2215455/8, 3088737, 428675377/16, 0, ...]; ...
%e A305139 in which the main diagonal is all zeros after the initial term, illustrating that 0 = [x^n] exp(n^2*Integral A(x)^2 dx)/A(x), for n > 0.
%e A305139 RELATED SERIES.
%e A305139 A(x)^2 = 1 + 2*x + 19*x^2 + 328*x^3 + 8001*x^4 + 247664*x^5 + 9188337*x^6 + 394725252*x^7 + 19194243265*x^8 + 1039762257722*x^9 + ...
%e A305139 exp( Integral A(x)^2 dx) = 1 + x + 3*x^2/2! + 45*x^3/3! + 2145*x^4/4! + 203085*x^5/5! + 30980475*x^6/6! + 6838973145*x^7/7! + 2045481775425*x^8/8! + 792696897387225*x^9/9! + ...
%e A305139 A'(x)/A(x) = 1 + 17*x + 439*x^2 + 14473*x^3 + 568296*x^4 + 25625759*x^5 + 1297831032*x^6 + 72732570537*x^7 + 4462331350255*x^8 + ...
%o A305139 (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)^2)) / Ser(A) )[m+1] );A[n+1]}
%o A305139 for(n=0,20,print1(a(n),", "))
%Y A305139 Cf. A305137, A305138, A305140, A305141, A305142, A305143.
%K A305139 nonn
%O A305139 0,3
%A A305139 _Paul D. Hanna_, May 31 2018