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.

A379699 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x) / (1 + 2*x*exp(x)) ).

Original entry on oeis.org

1, 3, 27, 430, 10013, 309146, 11932303, 553740958, 30053978361, 1868855561938, 131048808861491, 10232894227577462, 880688476587319573, 82836081054992159194, 8454536286883278469431, 930656450818831650930766, 109910217042754940709650417, 13862746369348872200671305890
Offset: 0

Views

Author

Seiichi Manyama, Dec 30 2024

Keywords

Crossrefs

Cf. A201470, A377373.
Cf. A379456, A379700.

Programs

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

Formula

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

A379700 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x) / (1 + 3*x*exp(x)) ).

Original entry on oeis.org

1, 4, 45, 889, 25613, 977271, 46586281, 2669066695, 178800270009, 13720918482235, 1187217400278941, 114379273346497611, 12144899525595998821, 1409261040317551952935, 177436990866294436727409, 24094164269339964351367231, 3510079015962779901366174449, 546103202348225740056486964467
Offset: 0

Views

Author

Seiichi Manyama, Dec 30 2024

Keywords

Crossrefs

Cf. A379456, A379699.
Cf. A377374.

Programs

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

Formula

a(n) = (n!/(n+1)) * Sum_{k=0..n} 3^(n-k) * (2*n-k+1)^k * binomial(n+1,n-k)/k!.

A379846 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x) / (1 + x*exp(2*x)) ).

Original entry on oeis.org

1, 2, 15, 202, 3993, 104896, 3449431, 136490768, 6319722513, 335372124160, 20074806151551, 1338341234648320, 98356732036224745, 7900673166769620992, 688709957632464564231, 64754459774124307019776, 6532479591772426224737697, 703834470938326183482621952
Offset: 0

Views

Author

Seiichi Manyama, Jan 04 2025

Keywords

Crossrefs

Cf. A088690, A379456, A379847.
Cf. A366232, A379701.

Programs

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

Formula

a(n) = (n!/(n+1)) * Sum_{k=0..n} (3*n-2*k+1)^k * binomial(n+1,n-k)/k!.
E.g.f. A(x) satisfies A(x) = exp(x*A(x)) / ( 1 - x*exp(3*x*A(x)) ). - Seiichi Manyama, Feb 04 2025

A379847 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x) / (1 + x*exp(3*x)) ).

Original entry on oeis.org

1, 2, 17, 259, 5773, 171021, 6342937, 283094309, 14785425081, 885090944809, 59765476266061, 4494836808752049, 372655043070926821, 33769844474642217293, 3320996349535681398849, 352267766021524028011981, 40091829710459334010532593, 4873329774181782935197522641
Offset: 0

Views

Author

Seiichi Manyama, Jan 04 2025

Keywords

Crossrefs

Cf. A088690, A379456, A379846.
Cf. A366233, A379702.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (4*n-3*k+1)^k*binomial(n+1, n-k)/k!)/(n+1);

Formula

a(n) = (n!/(n+1)) * Sum_{k=0..n} (4*n-3*k+1)^k * binomial(n+1,n-k)/k!.
E.g.f. A(x) satisfies A(x) = exp(x*A(x)) / ( 1 - x*exp(4*x*A(x)) ). - Seiichi Manyama, Feb 04 2025

A379862 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x)/(1 + x*exp(x))^2 ).

Original entry on oeis.org

1, 3, 29, 502, 12761, 430986, 18217813, 926514058, 55133781809, 3760088111938, 289240874117981, 24780044801646762, 2340229465310736073, 241563626661550193794, 27059024800372108029221, 3269263894468329061597546, 423798837014001794141132897, 58674726188995774863597090690
Offset: 0

Views

Author

Seiichi Manyama, Jan 04 2025

Keywords

Links

Crossrefs

Cf. A377553, A379861.
Cf. A379456.

Programs

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

Formula

E.g.f. A(x) satisfies A(x) = exp(x*A) * (1 + x * A(x) * exp(x*A(x)))^2.
a(n) = (n!/(n+1)) * Sum_{k=0..n} (n+k+1)^(n-k) * binomial(2*n+2,k)/(n-k)!.
Showing 1-5 of 5 results.

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

Content is available under The OEIS End-User License Agreement.