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A307401
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G.f. A(x) satisfies: A(x) = 1 + Sum_{k>=1} k*x^k*A(x)^k/(1 + x^k*A(x)^k).
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3
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1, 1, 2, 8, 26, 92, 360, 1416, 5698, 23513, 98346, 416418, 1783144, 7704322, 33546344, 147071592, 648636050, 2875822121, 12810531924, 57306505152, 257330920910, 1159517118330, 5241137123470, 23758569938458, 107983949179512, 491985193384077, 2246564114646650
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OFFSET
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0,3
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LINKS
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FORMULA
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G.f. A(x) satisfies: A(x) = (23 + theta_2(x*A(x))^4 + theta_3(x*A(x))^4)/24.
G.f. A(x) satisfies: A(x) = 1 + Sum_{k>=1} A000593(k)*x^k*A(x)^k.
G.f.: A(x) = (1/x)*Series_Reversion(x/(1 + Sum_{k>=1} A000593(k)*x^k)).
a(n) ~ c * d^n / n^(3/2), where d = 4.83361837854808845493127190842423391826598301272368919050344408629988519... and c = 0.506244425594072156224012562189085656331596921281799036166665... - Vaclav Kotesovec, Sep 27 2023
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EXAMPLE
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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 + ...
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MATHEMATICA
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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]
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]
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]
(* 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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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