cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A192206 G.f. satisfies: A(x) = 1 + Sum_{n>=1} x^n*A(x)^n/(1 - x^n*A(x)^n).

Original entry on oeis.org

1, 1, 3, 9, 32, 118, 460, 1844, 7587, 31804, 135433, 584092, 2546250, 11201310, 49663816, 221701658, 995621590, 4494862920, 20388491423, 92872814115, 424665159560, 1948516758192, 8968647197842, 41399782218408, 191608577837136
Offset: 0

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Author

Paul D. Hanna, Jun 25 2011

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 9*x^3 + 32*x^4 + 118*x^5 + 460*x^6 +...
which satisfies:
A(x) = 1 + x*A(x)/(1-x*A(x)) + x^2*A(x)^2/(1-x^2*A(x)^2) + x^3*A(x)^3/(1-x^3*A(x)^3) +...

Links

Crossrefs

Cf. A000005, A307397, A307399.

Programs

  • Mathematica
    nmax = 30; A[] = 0; Do[A[x] = 1 + Sum[x^k*A[x]^k/(1 - x^k*A[x]^k), {k, 1, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] (* Vaclav Kotesovec, Sep 28 2023 *)
(* Calculation of constant d: *) val = r /. FindRoot[{Log[1 - r*s] + QPolyGamma[0, 1, r*s] == (s-1)* Log[r*s], r*(1/(-1 + r*s) + Derivative[0, 0, 1][QPolyGamma][0, 1, r*s]) == Log[r*s] + (s-1)/s}, {r, 1/5}, {s, 2}, WorkingPrecision -> 30] // Quiet; N[1/Chop[val], -Floor[Log[10, Abs[Im[val]]]] - 3] (* Vaclav Kotesovec, Sep 28 2023 *)
  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, x^m*A^m/(1-x^m*A^m+x*O(x^n)))); polcoeff(A, n)}
  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, sigma(m,0)*x^m*A^m+x*O(x^n))); polcoeff(A, n)}

Formula

G.f. satisfies: A(x) = 1 + Sum_{n>=1} tau(n)*x^n*A(x)^n, where tau(n) = the number of divisors of n (A000005).
G.f. satisfies: G(x) = A(x/G(x)) where G(x) = 1 + Sum_{n>=1} x^n/(1 - x^n) is a g.f. for A000005.
a(n) ~ c * d^n / n^(3/2), where d = 4.92231207163377242376... and c = 0.58002108600295682... - Vaclav Kotesovec, Sep 28 2023

A307399 G.f. A(x) satisfies: A(x) = 1 + Sum_{k>=1} k*x^k*A(x)^k/(1 - x^k*A(x)^k).

Original entry on oeis.org

1, 1, 4, 14, 60, 262, 1218, 5798, 28364, 141239, 714532, 3660098, 18949830, 98997082, 521218206, 2762807736, 14731968812, 78968221213, 425282844540, 2299997984844, 12485972925500, 68015653648096, 371666798915578, 2036765196573550, 11190993772943502, 61637787236407747
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 07 2019

Keywords

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 14*x^3 + 60*x^4 + 262*x^5 + 1218*x^6 + 5798*x^7 + 28364*x^8 + 141239*x^9 + 714532*x^10 + ...

Links

Crossrefs

Cf. A000203, A192206, A307397, A307401.

Programs

  • Mathematica
    terms = 26; 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 = 26; A[] = 0; Do[A[x] = 1 + Sum[DivisorSigma[1, k] x^k A[x]^k, {k, 1, j}] + O[x]^j, {j, 1, terms}]; CoefficientList[A[x], x]
terms = 26; CoefficientList[1/x InverseSeries[Series[x/(1 + Sum[DivisorSigma[1, k]  x^k, {k, 1, terms}]), {x, 0, terms}], x], x]
(* Calculation of constant d: *) val = r /. FindRoot[{1 + (Log[1 - r*s] + QPolyGamma[0, 1, r*s]) / Log[r*s] == s + r*s*Derivative[0, 1][QPochhammer][r*s, r*s]/ QPochhammer[r*s, r*s], r^2*s*Derivative[0, 1][QPochhammer][r*s, r*s]^2/ QPochhammer[r*s, r*s]^2 + 1/(s*(-1 + r*s)*Log[r*s]^2) * (2*r*s*Log[r*s] + s*Log[r*s]^2 - r*s^2*Log[r*s]^2 + 2*Log[1 - r*s] - 2*r*s*Log[1 - r*s] + (2 - 2*r*s)*QPolyGamma[0, 1, r*s] + (1 - r*s)*QPolyGamma[1, 1, r*s] - 2*r*s*Log[r*s] * Derivative[0, 0, 1][QPolyGamma][0, 1, r*s] + 2*r^2*s^2*Log[r*s]*Derivative[0, 0, 1][QPolyGamma][0, 1, r*s]) == (r*(Derivative[0, 1][QPochhammer][r*s, r*s] + r*s*Derivative[0, 2][QPochhammer][r*s, r*s]))/QPochhammer[r*s, r*s]}, {r, 1/6}, {s, 2}, WorkingPrecision -> 40] // Quiet; N[ 1/Chop[val], -Floor[Log[10, Abs[Im[val]]]] - 3] (* Vaclav Kotesovec, Oct 02 2023 *)

Formula

G.f. A(x) satisfies: A(x) = 1 + Sum_{k>=1} sigma(k)*x^k*A(x)^k.
G.f.: A(x) = (1/x)*Series_Reversion(x/(1 + Sum_{k>=1} sigma(k)*x^k)).
a(n) ~ c * d^n / n^(3/2), where d = A366072 = 5.84278321476352032847350429253643509033417800773284061845774243558820314... and c = 0.5552806478004840811027181339325620905324642078294... - Vaclav Kotesovec, Apr 07 2019

A307401 G.f. A(x) satisfies: A(x) = 1 + Sum_{k>=1} k*x^k*A(x)^k/(1 + x^k*A(x)^k).

Original entry on oeis.org

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
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 07 2019

Keywords

Examples

			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 + ...

Crossrefs

Cf. A000593, A190790, A192206, A307397, A307399.

Programs

  • Mathematica
    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 *)

Formula

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
Showing 1-3 of 3 results.

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