The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A307401

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A307401

G.f. A(x) satisfies: A(x) = 1 + Sum_{k>=1} k*x^k*A(x)^k/(1 + x^k*A(x)^k).
(history; published version)

#14 by Vaclav Kotesovec at Wed Sep 27 12:39:11 EDT 2023

STATUS

editing

approved

#13 by Vaclav Kotesovec at Wed Sep 27 12:39:04 EDT 2023

MATHEMATICA

(* 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 *)

#12 by Vaclav Kotesovec at Wed Sep 27 12:38:15 EDT 2023

MATHEMATICA

(* 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 *)

STATUS

approved

editing

#11 by Vaclav Kotesovec at Wed Sep 27 12:35:23 EDT 2023

STATUS

editing

approved

#10 by Vaclav Kotesovec at Wed Sep 27 12:35:11 EDT 2023

MATHEMATICA

(* Calculation of constants {d, c} : *) {1/r, Sqrt[63*s/(2*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 *)

STATUS

approved

editing

#9 by Vaclav Kotesovec at Wed Sep 27 12:33:24 EDT 2023

STATUS

editing

approved

#8 by Vaclav Kotesovec at Wed Sep 27 12:33:00 EDT 2023

MATHEMATICA

(* Calculation of constant constants {d, c} : *) {1/r, Sqrt[6*s/(2*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 *)

#7 by Vaclav Kotesovec at Wed Sep 27 12:17:20 EDT 2023

MATHEMATICA

(* Calculation of constant d: *) 1/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 *)

#6 by Vaclav Kotesovec at Wed Sep 27 12:16:39 EDT 2023

FORMULA

a(n) ~ c * d^n / n^(3/2), where d = 4.83361837854808845493127190842423391826598301272368919050344408629988519... and c = 0.506244425594072156224012562189085656331596921281799036166665... - Vaclav Kotesovec, Sep 27 2023

STATUS

approved

editing

#5 by Susanna Cuyler at Sun Apr 07 09:08:02 EDT 2019

STATUS

proposed

approved