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%I #14 Sep 27 2023 12:39:11
%S 1,1,2,8,26,92,360,1416,5698,23513,98346,416418,1783144,7704322,
%T 33546344,147071592,648636050,2875822121,12810531924,57306505152,
%U 257330920910,1159517118330,5241137123470,23758569938458,107983949179512,491985193384077,2246564114646650
%N G.f. A(x) satisfies: A(x) = 1 + Sum_{k>=1} k*x^k*A(x)^k/(1 + x^k*A(x)^k).
%F G.f. A(x) satisfies: A(x) = (23 + theta_2(x*A(x))^4 + theta_3(x*A(x))^4)/24.
%F G.f. A(x) satisfies: A(x) = 1 + Sum_{k>=1} A000593(k)*x^k*A(x)^k.
%F G.f.: A(x) = (1/x)*Series_Reversion(x/(1 + Sum_{k>=1} A000593(k)*x^k)).
%F a(n) ~ c * d^n / n^(3/2), where d = 4.83361837854808845493127190842423391826598301272368919050344408629988519... and c = 0.506244425594072156224012562189085656331596921281799036166665... - _Vaclav Kotesovec_, Sep 27 2023
%e G.f.: A(x) = 1 + x + 2*x^2 + 8*x^3 + 26*x^4 + 92*x^5 + 360*x^6 + 1416*x^7 + 5698*x^8 + 23513*x^9 + 98346*x^10 + ...
%t terms = 27; A[_] = 0; Do[A[x_] = 1 + Sum[k x^k A[x]^k/(1 + x^k A[x]^k), {k, 1, j}] + O[x]^j, {j, 1, terms}]; CoefficientList[A[x], x]
%t terms = 27; A[_] = 0; Do[A[x_] = 1 + Sum[Sum[(-1)^(k/d + 1) d, {d, Divisors[k]}] x^k A[x]^k, {k, 1, j}] + O[x]^j, {j, 1, terms}]; CoefficientList[A[x], x]
%t terms = 27; CoefficientList[1/x InverseSeries[Series[x/(1 + Sum[Sum[(-1)^(k/d + 1) d, {d, Divisors[k]}] x^k, {k, 1, terms}]), {x, 0, terms}], x], x]
%t (* Calculation of constants {d, c} : *) {1/r, Sqrt[3*s/(Pi*(3*EllipticTheta[2, 0, r*s]^2 * Derivative[0, 0, 1][EllipticTheta][2, 0, r*s]^2 + 3*EllipticTheta[3, 0, r*s]^2 * Derivative[0, 0, 1][EllipticTheta][3, 0, r*s]^2 + EllipticTheta[2, 0, r*s]^3 * Derivative[0, 0, 2][EllipticTheta][2, 0, r*s] + EllipticTheta[3, 0, r*s]^3 * Derivative[0, 0, 2][EllipticTheta][3, 0, r*s]))]/r} /. FindRoot[{24*s == 23 + EllipticTheta[2, 0, r*s]^4 + EllipticTheta[3, 0, r*s]^4, r*EllipticTheta[2, 0, r*s]^3 * Derivative[0, 0, 1][EllipticTheta][2, 0, r*s] + r*EllipticTheta[3, 0, r*s]^3 * Derivative[0, 0, 1][EllipticTheta][3, 0, r*s] == 6}, {r, 1/3}, {s, 1}, WorkingPrecision -> 120] (* _Vaclav Kotesovec_, Sep 27 2023 *)
%Y Cf. A000593, A190790, A192206, A307397, A307399.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Apr 07 2019
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