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.

A373526 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, -22679, 703081, 9397081, 59946481, 510926131, -1770628859, 28435376971, 879567081121, 1118967005521, 133425791669521, -153681150137039, 1206979480409761, 974131304609110411, -2730300052811686739, 68047061689139820211
Offset: 0

Views

Author

Seiichi Manyama, Jun 08 2024

Keywords

Crossrefs

Cf. A190877, A373524, A373525.
Cf. A373521.

Programs

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

Formula

a(n) = n! * Sum_{k=0..floor(n/4)} binomial(3*n/4-3*k,k)/(n-4*k)!.
a(n) == 1 mod 90.

A373520 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, 143641, 1829521, 12501721, 59922721, 2173048021, 44315751481, 478799701381, 3492321094081, 116722067432881, 3290135175240481, 50242015215929521, 508061488330088641, 16418736123292904941, 585427887134915295241
Offset: 0

Views

Author

Seiichi Manyama, Jun 08 2024

Keywords

Crossrefs

Cf. A293507, A373519, A373521.
Cf. A373525.

Programs

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

Formula

a(n) = n! * Sum_{k=0..floor(n/4)} binomial(n/2-k-1,k)/(n-4*k)!.
a(n) == 1 mod 60.
From Vaclav Kotesovec, Sep 03 2025: (Start)
Recurrence: (n-8)*a(n) = (n-8)*a(n-2) + 3*(n-4)*(n-3)*(n-2)*(n^2 - 11*n + 20)*a(n-4) + 2*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*a(n-6) - 3*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n^2 - 13*n + 32)*a(n-8) + (n-9)*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)^2*(n-3)*(n-2)*a(n-10) + (n-12)*(n-11)^2*(n-10)*(n-9)*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)^2*(n-3)*(n-2)*a(n-12).
a(n) ~ 2^(-1/6) * 3^(-1/2) * exp(3*2^(-4/3)*n^(1/3) - n) * n^(n - 1/3).
(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.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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