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 1-4 of 4 results.

A373524 Expansion of e.g.f. exp(x * (1 + x^4)^(1/4)).

Original entry on oeis.org

1, 1, 1, 1, 1, 31, 181, 631, 1681, -30239, -219239, -609839, 23761, 348686911, 2769787021, 8683865191, 519049441, -13487418759359, -125598814684559, -446790263130719, 15237796321, 1447967425273506271, 15403882866996490021, 61625443167286653271, 1295472977069041
Offset: 0

Views

Author

Seiichi Manyama, Jun 08 2024

Keywords

Crossrefs

Cf. A190877, A373525, A373526.
Cf. A373519.

Programs

  • PARI
    a(n) = n!*sum(k=0, n\4, binomial(n/4-k, k)/(n-4*k)!);

Formula

a(n) = n! * Sum_{k=0..floor(n/4)} binomial(n/4-k,k)/(n-4*k)!.
a(n) == 1 mod 30.

A373525 Expansion of e.g.f. exp(x * (1 + x^4)^(1/2)).

Original entry on oeis.org

1, 1, 1, 1, 1, 61, 361, 1261, 3361, -37799, 15121, 2522521, 20005921, 486563221, 363363001, -12486293819, 3113772481, -12960051533519, 8909829442081, 528011239172401, 241435513794241, 1381310884267333741, 3097715443570441, -49080307628073705059
Offset: 0

Views

Author

Seiichi Manyama, Jun 08 2024

Keywords

Crossrefs

Cf. A190877, A373524, A373526.
Cf. A373520.

Programs

  • PARI
    a(n) = n!*sum(k=0, n\4, binomial(n/2-2*k, k)/(n-4*k)!);

Formula

a(n) = n! * Sum_{k=0..floor(n/4)} binomial(n/2-2*k,k)/(n-4*k)!.
a(n) == 1 mod 60.

A373521 Expansion of e.g.f. exp(x/(1 - x^4)^(3/4)).

Original entry on oeis.org

1, 1, 1, 1, 1, 91, 541, 1891, 5041, 249481, 3424681, 24365881, 119821681, 4208219731, 96849813061, 1133147785771, 8728726799521, 251218306095121, 8116398738418321, 138787857114672241, 1523943014238675361, 39648007379230971211, 1599866285860593980461
Offset: 0

Views

Author

Seiichi Manyama, Jun 08 2024

Keywords

Crossrefs

Cf. A293507, A373519, A373520.
Cf. A373526.

Programs

  • Mathematica
    nmax = 25; CoefficientList[Series[E^(x/(1 - x^4)^(3/4)), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Sep 03 2025 *)
  • PARI
    a(n) = n!*sum(k=0, n\4, binomial(3*n/4-2*k-1, k)/(n-4*k)!);

Formula

a(n) = n! * Sum_{k=0..floor(n/4)} binomial(3*n/4-2*k-1,k)/(n-4*k)!.
a(n) == 1 mod 90.
From Vaclav Kotesovec, Sep 03 2025: (Start)
a(n) ~ 3^(2/7) * 7^(-1/2) * exp(7*3^(-3/7)*n^(3/7)/4 - n) * n^(n - 2/7) * (1 + 3^(1/7)/(8*n^(1/7))).
(End)

A373708 Expansion of e.g.f. exp(x * (1 + x^4)^2).

Original entry on oeis.org

1, 1, 1, 1, 1, 241, 1441, 5041, 13441, 393121, 10946881, 99902881, 559025281, 2335441681, 182348406241, 4382526067921, 48882114328321, 355837396998721, 5157802930734721, 312898934463543361, 7129755898022511361, 89524038506304371761, 773103613914955683361
Offset: 0

Views

Author

Seiichi Manyama, Jun 14 2024

Keywords

Crossrefs

Cf. A361278, A373578, A373707.
Cf. A190877, A373524, A373525, A373526.
Cf. A373706.

Programs

  • PARI
    a(n) = n!*sum(k=0, 2*n\9, binomial(2*n-8*k, k)/(n-4*k)!);

Formula

a(n) = n! * Sum_{k=0..floor(2*n/9)} binomial(2*n-8*k,k)/(n-4*k)!.
a(n) == 1 (mod 240).
a(n) = a(n-1) + 10*(n-1)*(n-2)*(n-3)*(n-4)*a(n-5) + 9*(n-1)*(n-2)*(n-3)*(n-4)*(n-5)*(n-6)*(n-7)*(n-8)*a(n-9).
Showing 1-4 of 4 results.

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