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 1-7 of 7 results.

A377893 E.g.f. A(x) satisfies A(x) = (1 + x*A(x)^3) * exp(x * A(x)^3).

Original entry on oeis.org

1, 2, 27, 742, 31337, 1796376, 130408603, 11472417104, 1186462228785, 141083381264896, 18966727953873371, 2844742575536036352, 470958524169176911513, 85307709328403287961600, 16782586179544965856158363, 3563492814539574559964993536, 812273035493592514001487147233
Offset: 0

Views

Author

Seiichi Manyama, Nov 11 2024

Keywords

Crossrefs

Cf. A088690, A377892.

Programs

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

Formula

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

A379867 E.g.f. A(x) satisfies A(x) = 1/(exp(-x*A(x)^2) - x*A(x)^2).

Original entry on oeis.org

1, 2, 23, 529, 18589, 884281, 53195407, 3874595089, 331580316473, 32614443047521, 3625839880813171, 449629404853604185, 61535275741655857621, 9213155228282408405185, 1498018121369750569371959, 262869047482982449625840161, 49515850496472530668242845041
Offset: 0

Views

Author

Seiichi Manyama, Jan 04 2025

Keywords

Crossrefs

Cf. A377890, A379870.
Cf. A377892, A379884.
Cf. A377891, A377893, A379886.
Cf. A364985, A379864.

Programs

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

Formula

E.g.f.: sqrt( (1/x) * Series_Reversion( x * (exp(-x) - x)^2 ) ).
a(n) = n! * Sum_{k=0..n} (3*n-k+1)^(k-1) * binomial(3*n-k+1,n-k)/k!.

A379884 E.g.f. A(x) satisfies A(x) = 1/(exp(-x*A(x)^2) - x).

Original entry on oeis.org

1, 2, 15, 223, 5045, 154161, 5949715, 277816813, 15234148585, 959821848433, 68333878996991, 5425649143910733, 475370226250388221, 45559752911807595865, 4741534923025152367627, 532526268840445510805341, 64198018232238090097818065, 8268729272698380485865553761
Offset: 0

Views

Author

Seiichi Manyama, Jan 05 2025

Keywords

Crossrefs

Cf. A377892, A379867.
Cf. A088690.

Programs

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

Formula

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

A380646 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-2*x)/(1 + x)^2 ).

Original entry on oeis.org

1, 4, 46, 932, 27568, 1080432, 52916176, 3115326496, 214470890496, 16914853191680, 1504252282653184, 148956086481767424, 16256865070022066176, 1938988214539948730368, 250943399365390735104000, 35026523834624205803491328, 5245178283068781060488298496, 838841884254236846183525646336
Offset: 0

Views

Author

Seiichi Manyama, Feb 06 2025

Keywords

Links

Crossrefs

Cf. A088690, A380647, A380648.
Cf. A065866, A377829.
Cf. A097629, A380828.
Cf. A377892.

Programs

  • Mathematica
    nmax=18; CoefficientList[(1/x)InverseSeries[Series[x*Exp[-2*x]/(1 + x)^2 ,{x,0,nmax}]],x]Range[0,nmax-1]! (* Stefano Spezia, Feb 06 2025 *)
  • PARI
    a(n) = 2*n!*sum(k=0, n, (2*n+2)^(k-1)*binomial(2*n+2, n-k)/k!);

Formula

E.g.f. A(x) satisfies A(x) = (1 + x*A(x))^2 * exp(2 * x * A(x)).
E.g.f.: B(x)^2, where B(x) is the e.g.f. of A377892.
a(n) = 2 * n! * Sum_{k=0..n} (2*n+2)^(k-1) * binomial(2*n+2,n-k)/k!.

A379885 E.g.f. A(x) satisfies A(x) = 1/(exp(-x) - x*A(x)).

Original entry on oeis.org

1, 2, 11, 118, 1885, 40266, 1080679, 34979134, 1326825497, 57744176914, 2836795756771, 155305155441030, 9376803979425205, 619006372481008474, 44357422104298022399, 3429215554499681260366, 284496868838293052890033, 25212167721275946619910178, 2377021703587467346833760315
Offset: 0

Views

Author

Seiichi Manyama, Jan 05 2025

Keywords

Crossrefs

Cf. A377890, A377892.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(2/(exp(-x)+sqrt(exp(-2*x)-4*x))))
  • PARI
    a(n) = n!*sum(k=0, n, (2*n-2*k+1)^(k-1)*binomial(2*n-2*k+1, n-k)/k!);

Formula

E.g.f.: 2/(exp(-x) + sqrt(exp(-2*x) - 4*x)).
a(n) = n! * Sum_{k=0..n} (2*n-2*k+1)^(k-1) * binomial(2*n-2*k+1,n-k)/k!.
a(n) ~ sqrt(1 + LambertW(1/2)) * 2^n * n^(n-1) / (LambertW(1/2)^(n + 1/2) * exp(n)). - Vaclav Kotesovec, Jan 05 2025

A379912 E.g.f. A(x) satisfies A(x) = 1/(exp(-x*A(x)^3) - x*A(x)).

Original entry on oeis.org

1, 2, 23, 541, 19585, 962901, 59969227, 4526706661, 401724516641, 40994441922169, 4729721311570411, 608827327842480825, 86507217246635276065, 13448830748996370988885, 2270847762050485928361227, 413849998079530364443224781, 80967576778854924208520130241
Offset: 0

Views

Author

Seiichi Manyama, Jan 05 2025

Keywords

Links

Crossrefs

Cf. A377893, A379870, A379897.
Cf. A377890, A377892, A379885.

Programs

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

Formula

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

A379940 E.g.f. A(x) satisfies A(x) = 1/( exp(-x*A(x)^(2/3)) - x*A(x)^(1/3) )^3.

Original entry on oeis.org

1, 6, 81, 1788, 55785, 2267298, 114015825, 6848257272, 478929874257, 38253577287870, 3437561332041969, 343381977748134900, 37755068758105209849, 4531920849132497127258, 589779214651388664049905, 82722149483353129407482352, 12440903535778778244423710625, 1997259670949248788135594940278
Offset: 0

Views

Author

Seiichi Manyama, Jan 07 2025

Keywords

Crossrefs

Cf. A377892, A379936.

Programs

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

Formula

E.g.f.: B(x)^3, where B(x) is the e.g.f. of A377892.
a(n) = 3 * n! * Sum_{k=0..n} (2*n+3)^(k-1) * binomial(2*n+3,n-k)/k!.
a(n) == 0 (mod 3) for n>0.
Showing 1-7 of 7 results.

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

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