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.

A380014 Expansion of e.g.f. 1/sqrt(exp(-2*x) - 2*x).

Original entry on oeis.org

1, 2, 10, 88, 1084, 17176, 332824, 7623904, 201540112, 6038820640, 202246657696, 7486877795200, 303561658686400, 13378863292503424, 636833910410881408, 32559375816074384896, 1779494669204225605888, 103532173699456380625408, 6388705590982575700625920
Offset: 0

Views

Author

Seiichi Manyama, Jan 09 2025

Keywords

Crossrefs

Cf. A072597, A380016, A380018.

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) ~ 2^(n + 1/2) * n^n / (sqrt(1 + LambertW(1)) * LambertW(1)^(n + 1/2) * exp(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)!.

A380156 Expansion of e.g.f. 1/(1 - 3*x*exp(3*x))^(1/3).

Original entry on oeis.org

1, 1, 10, 127, 2260, 52165, 1478098, 49666267, 1930817080, 85253566825, 4214519350750, 230609701370719, 13837049296702228, 903380930924784013, 63754235596937808874, 4836352735401636409795, 392451456493513697671792, 33920902255644870783973201, 3111255003645991777552833718
Offset: 0

Views

Author

Seiichi Manyama, Jan 13 2025

Keywords

Crossrefs

Cf. A006153, A380155.
Cf. A380016, A380017.

Programs

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

Formula

a(n) = 3^n * n! * Sum_{k=0..n} (-1)^k * k^(n-k) * binomial(-1/3,k)/(n-k)!.
a(n) == 1 (mod 9).

A380021 Expansion of e.g.f. 1/(exp(-3*x) - 3*x*exp(-2*x))^(1/3).

Original entry on oeis.org

1, 2, 9, 77, 977, 16281, 335173, 8208901, 233037185, 7522621505, 272096862821, 10899761462085, 478990330829233, 22910468287983121, 1184832950732237381, 65877062190857942981, 3918656527419803705729, 248317978064709144523521, 16699787528059828201246021
Offset: 0

Views

Author

Seiichi Manyama, Jan 09 2025

Keywords

Crossrefs

Cf. A380016, A380017.

Programs

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

Formula

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

A380159 Expansion of e.g.f. (exp(-3*x) + 3*x)^(1/3).

Original entry on oeis.org

1, 0, 3, -9, -27, 459, 243, -58563, 338985, 11581623, -206336889, -2610099207, 128764066797, 37135699587, -90848500643781, 1216300295221749, 68623945856512209, -2410073970973057809, -44786917868989757553, 4171855691698864732305, -8174731579262161250859
Offset: 0

Views

Author

Seiichi Manyama, Jan 13 2025

Keywords

Crossrefs

Cf. A380016, A380158.

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)!.
a(n) == 0 (mod 3) for n>0.
Showing 1-5 of 5 results.

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

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