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 1-5 of 5 results.

A380016 Expansion of e.g.f. 1/(exp(-3*x) - 3*x)^(1/3).

Original entry on oeis.org

1, 2, 13, 161, 2833, 64841, 1827685, 61192181, 2372620801, 104549934977, 5160225776101, 281994042839477, 16902276273364465, 1102519010117525105, 77749077431938305541, 5894145002422856684501, 478015727336387513545345, 41295912476641866286397825, 3786025873450493919700627525
Offset: 0

Views

Author

Seiichi Manyama, Jan 09 2025

Keywords

Crossrefs

Cf. A072597, A380014, A380018.

Programs

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

Formula

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

A380155 Expansion of e.g.f. 1/sqrt(1 - 2*x*exp(2*x)).

Original entry on oeis.org

1, 1, 7, 63, 785, 12545, 244407, 5619775, 148977313, 4473497601, 150078670055, 5563415292479, 225832882678449, 9962766560986369, 474619650950131351, 24283168467229957695, 1327993894505461755713, 77305844496338607597569, 4772660185400974888323015
Offset: 0

Views

Author

Seiichi Manyama, Jan 13 2025

Keywords

Crossrefs

Cf. A006153, A380156.
Cf. A380014, A380015.

Programs

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

Formula

a(n) = 2^n * n! * Sum_{k=0..n} (-1)^k * k^(n-k) * binomial(-1/2,k)/(n-k)!.
a(n) == 1 (mod 2).
a(n) ~ 2^(n + 1/2) * n^n / (sqrt(1 + LambertW(1)) * exp(n) * LambertW(1)^n). - Vaclav Kotesovec, Jan 23 2025

A380018 Expansion of e.g.f. 1/(exp(-4*x) - 4*x)^(1/4).

Original entry on oeis.org

1, 2, 16, 256, 5856, 175296, 6486016, 285756416, 14606007296, 849615763456, 55415153442816, 4005309938466816, 317750919017168896, 27449350209163821056, 2564871898004949303296, 257753802183061443444736, 27720748513211258671988736, 3176821722223524679312736256
Offset: 0

Views

Author

Seiichi Manyama, Jan 09 2025

Keywords

Crossrefs

Cf. A072597, A380014, A380016.

Programs

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

Formula

a(n) = n! * Sum_{k=0..n} (-4)^k * (4*k+1)^(n-k) * binomial(-1/4,k)/(n-k)!.

A380158 Expansion of e.g.f. sqrt(exp(-2*x) + 2*x).

Original entry on oeis.org

1, 0, 2, -4, -4, 64, -8, -3312, 14352, 267776, -3403744, -24119360, 881205184, -593040384, -261913919616, 2567414468864, 83291021050112, -2080429273726976, -22004502593928704, 1526354137528335360, -3870482611349750784, -1112746657730132623360, 18568218633016319670272
Offset: 0

Views

Author

Seiichi Manyama, Jan 13 2025

Keywords

Crossrefs

Cf. A191498, A380014.

Programs

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

Formula

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

A380020 Expansion of e.g.f. 1/sqrt(exp(-2*x) - 2*x*exp(-x)).

Original entry on oeis.org

1, 2, 8, 55, 540, 6861, 106828, 1968443, 41884496, 1010558161, 27259824996, 812935829355, 26556802948624, 943118750625377, 36176486632451012, 1490585029223430691, 65656827447552549504, 3078782615385684631809, 153127047650469476373316
Offset: 0

Views

Author

Seiichi Manyama, Jan 09 2025

Keywords

Crossrefs

Cf. A380014, A380015.

Programs

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

Formula

a(n) = n! * Sum_{k=0..n} (-2)^k * (k+1)^(n-k) * binomial(-1/2,k)/(n-k)!.
Showing 1-5 of 5 results.

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

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