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 1-10 of 11 results. Next

A364987 E.g.f. satisfies A(x) = 1 + x*exp(x)*A(x)^4.

Original entry on oeis.org

1, 1, 10, 183, 5140, 196005, 9468486, 554425963, 38171336680, 3022130473065, 270537702834250, 27021535857472431, 2979254055371578524, 359411244032212931533, 47093111659782104431438, 6660135357832421444841555, 1011181346455643980818939856
Offset: 0

Views

Author

Seiichi Manyama, Aug 15 2023

Keywords

Crossrefs

Cf. A006153, A295238, A364983.
Cf. A002293, A349331.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, k^(n-k)*binomial(4*k, k)/((3*k+1)*(n-k)!));

Formula

a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(4*k+1,k)/( (4*k+1)*(n-k)! ) = n! * Sum_{k=0..n} k^(n-k) * A002293(k)/(n-k)!.
a(n) ~ 2*sqrt(1 + LambertW(27/256)) * n^(n-1) / (3^(3/2) * exp(n) * LambertW(27/256)^n). - Vaclav Kotesovec, Nov 11 2024

A364985 E.g.f. satisfies A(x) = 1 + x*A(x)^3*exp(x*A(x)^2).

Original entry on oeis.org

1, 1, 8, 123, 2884, 91445, 3664926, 177796759, 10132646840, 663644108169, 49123993335130, 4055804550134051, 369544757016476196, 36834870020525413213, 3987179241476814768854, 465777171342934543710255, 58407238852473276959363056, 7825395596421876706944643985
Offset: 0

Views

Author

Seiichi Manyama, Aug 15 2023

Keywords

Crossrefs

Cf. A364983, A364984, A364986.
Cf. A003168.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, k^(n-k)*binomial(2*n+k+1, k)/((2*n+k+1)*(n-k)!));

Formula

a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(2*n+k+1,k)/( (2*n+k+1)*(n-k)! ).

A364984 E.g.f. satisfies A(x) = 1 + x*A(x)^3*exp(x*A(x)).

Original entry on oeis.org

1, 1, 8, 117, 2596, 77705, 2936406, 134228059, 7204913528, 444331053873, 30963240318250, 2406301353714731, 206354828717754036, 19357367027097743449, 1971809610601104110942, 216754216326949771274715, 25575749384428387961718256, 3224227609551980271408565985
Offset: 0

Views

Author

Seiichi Manyama, Aug 15 2023

Keywords

Crossrefs

Cf. A364983, A364985, A364986.
Cf. A108447.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, k^(n-k)*binomial(n+2*k+1, k)/((n+2*k+1)*(n-k)!));

Formula

a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(n+2*k+1,k)/( (n+2*k+1)*(n-k)! ).

A364986 E.g.f. satisfies A(x) = 1 + x*A(x)^3*exp(x*A(x)^3).

Original entry on oeis.org

1, 1, 8, 129, 3196, 107465, 4575966, 236120059, 14322901832, 998966928897, 78770826493210, 6929685905371691, 672900446143476156, 71491442785783506577, 8249400210035835040022, 1027394346436911560475915, 137360293432089585554830096
Offset: 0

Views

Author

Seiichi Manyama, Aug 15 2023

Keywords

Crossrefs

Cf. A364983, A364984, A364985.
Cf. A161633, A364982, A364989.
Cf. A002293.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, k^(n-k)*binomial(3*n+1, k)/(n-k)!)/(3*n+1);

Formula

a(n) = (n!/(3*n+1)) * Sum_{k=0..n} k^(n-k) * binomial(3*n+1,k)/(n-k)!.

A377503 E.g.f. satisfies A(x) = 1/(1 - x * exp(x) * A(x))^2.

Original entry on oeis.org

1, 2, 18, 270, 5936, 173330, 6335772, 278724362, 14350790064, 847007698338, 56397332340020, 4182866692785242, 342022887565717800, 30570009715185100082, 2965368922693150575084, 310276298423966343555690, 34834957115496822249510752, 4177193847524372747798263106
Offset: 0

Views

Author

Seiichi Manyama, Oct 30 2024

Keywords

Crossrefs

Cf. A295238, A377504, A377528.
Cf. A006013, A364983.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, k^(n-k)*binomial(3*k+1, k)/((k+1)*(n-k)!));

Formula

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A364983.
a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(3*k+1,k)/( (k+1)*(n-k)! ).

A377526 E.g.f. satisfies A(x) = 1 + x*exp(x)*A(x)^5.

Original entry on oeis.org

1, 1, 12, 273, 9604, 460105, 27966126, 2062219117, 178897527768, 17853102321489, 2014988044093210, 253792946798597701, 35290880970687039732, 5370055269772474994713, 887591963820839894529654, 158357028389450319651183165, 30332317748593431632078480176, 6208425034878692992471996557217
Offset: 0

Views

Author

Seiichi Manyama, Oct 30 2024

Keywords

Comments

In general, for k > 1, if e.g.f. satisfies A(x) = 1 + x*exp(x)*A(x)^k, then a(n) ~ sqrt(k*(1 + LambertW((k-1)^(k-1)/k^k))) * n^(n-1) / ((k-1)^(3/2) * exp(n) * LambertW((k-1)^(k-1)/k^k)^n). - Vaclav Kotesovec, Nov 11 2024

Crossrefs

Cf. A006153, A295238, A364983, A364987.
Cf. A002294.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, k^(n-k)*binomial(5*k, k)/((4*k+1)*(n-k)!));

Formula

a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(5*k,k)/( (4*k+1)*(n-k)! ) = n! * Sum_{k=0..n} k^(n-k) * A002294(k)/(n-k)!.
a(n) ~ sqrt(5*(1 + LambertW(256/3125))) * n^(n-1) / (8 * exp(n) * LambertW(256/3125)^n). - Vaclav Kotesovec, Nov 11 2024

A377575 E.g.f. satisfies A(x) = (1 + x * exp(x) * A(x))^3.

Original entry on oeis.org

1, 3, 30, 483, 11100, 334035, 12478698, 558058179, 29104042152, 1735547479587, 116539815603630, 8704631976941043, 716019297815418732, 64326542671867079955, 6267631435921525638738, 658359915933162131600355, 74168964857766293453918928, 8921104769819780822122624323
Offset: 0

Views

Author

Seiichi Manyama, Nov 02 2024

Keywords

Crossrefs

Cf. A006153, A377574, A377576.
Cf. A364983, A377554.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, k^(n-k)*binomial(3*k+3, k)/((k+1)*(n-k)!));

Formula

E.g.f.: B(x)^3, where B(x) is the e.g.f. of A364983.
a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(3*k+3,k)/( (k+1)*(n-k)! ).

A381999 E.g.f. A(x) satisfies A(x) = 1 + x*exp(2*x)*A(x)^3.

Original entry on oeis.org

1, 1, 10, 156, 3656, 115400, 4595232, 221281312, 12510826624, 812633118336, 59642105050880, 4881685773730304, 440905471531302912, 43559980305765793792, 4673231270870843441152, 541042726968231082967040, 67236501012517546330062848, 8927220151967826907452440576
Offset: 0

Views

Author

Seiichi Manyama, Mar 12 2025

Keywords

Crossrefs

Cf. A336950, A381997, A381998, A382000, A382001.
Cf. A001764, A364983, A371318.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (2*k)^(n-k)*binomial(3*k+1, k)/((3*k+1)*(n-k)!));

Formula

a(n) = n! * Sum_{k=0..n} (2*k)^(n-k) * A001764(k)/(n-k)!.
a(n) ~ sqrt(3*(1 + LambertW(8/27))) * 2^(n - 3/2) * n^(n-1) / (exp(n) * LambertW(8/27)^n). - Vaclav Kotesovec, Mar 22 2025

A365010 E.g.f. satisfies A(x) = 1 + x*exp(-x)*A(x)^3.

Original entry on oeis.org

1, 1, 4, 39, 596, 12365, 324714, 10329655, 386190328, 16597810233, 806356830230, 43700423019011, 2613919719004692, 171053575111641157, 12156558707970920866, 932424974682447304815, 76772968644326739801584, 6754080601542663692950769
Offset: 0

Views

Author

Seiichi Manyama, Aug 15 2023

Keywords

Crossrefs

Cf. A295239, A302397, A365011.
Cf. A001764, A364983.

Programs

  • Maple
    A365010 := proc(n)
    add( (-k)^(n-k)*A001764(k)/(n-k)!,k=0..n) ;
    %*n! ;
end proc:
seq(A365010(n),n=0..80); # R. J. Mathar, Aug 16 2023
  • PARI
    a(n) = n!*sum(k=0, n, (-k)^(n-k)*binomial(3*k, k)/((2*k+1)*(n-k)!));

Formula

a(n) = n! * Sum_{k=0..n} (-k)^(n-k) * A001764(k)/(n-k)!.

A377741 E.g.f. satisfies A(x) = exp(x) * (1 + x * A(x))^3.

Original entry on oeis.org

1, 4, 37, 583, 13225, 394681, 14659537, 652829857, 33937422001, 2018665692721, 135274646371561, 10087017309339433, 828563190097478425, 74348364577760978329, 7236649495742795579809, 759466703902106082652321, 85492204279344776678878945, 10275933748282019792253453025
Offset: 0

Views

Author

Seiichi Manyama, Nov 05 2024

Keywords

Crossrefs

Cf. A072597, A377740.
Cf. A364983, A377575.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (k+1)^(n-k-1)*binomial(3*k+3, k)/(n-k)!);

Formula

a(n) = n! * Sum_{k=0..n} (k+1)^(n-k-1) * binomial(3*k+3,k)/(n-k)!.
a(n) = A364983(n+1)/(n+1).
Showing 1-10 of 11 results. Next

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