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.

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A365175 E.g.f. satisfies A(x) = 1 + x*A(x)^4*exp(x*A(x)).

Original entry on oeis.org

1, 1, 10, 189, 5476, 215145, 10701006, 644909503, 45687408712, 3721382812305, 342689189598010, 35206864089944151, 3992473080042706524, 495361299387667990537, 66752437447119717428422, 9708649781691227748131535, 1515863453268825963300368656
Offset: 0

Views

Author

Seiichi Manyama, Aug 25 2023

Keywords

Crossrefs

Cf. A364987, A364989, A365176, A365177.
Cf. A161633, A213644, A364984.

Programs

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

Formula

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

A370927 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 - x*exp(x^2)) ).

Original entry on oeis.org

1, 1, 4, 36, 480, 8460, 187200, 4998000, 156387840, 5614313040, 227520921600, 10275211679040, 511772590264320, 27870149349282240, 1647541857684602880, 105073768465758892800, 7191330561736409088000, 525746801445336504633600
Offset: 0

Views

Author

Seiichi Manyama, Mar 06 2024

Keywords

Links

Crossrefs

Cf. A213644, A370928.

Programs

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

Formula

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

A370984 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 - x^2*exp(x)) ).

Original entry on oeis.org

1, 0, 2, 6, 84, 860, 14430, 257082, 5678456, 140241096, 3952791450, 123539438990, 4266378769092, 160943793753756, 6592371152535350, 291260465060881890, 13809548247503299440, 699362685890810753552, 37679514498664685654706
Offset: 0

Views

Author

Seiichi Manyama, Mar 06 2024

Keywords

Links

Crossrefs

Cf. A213644, A370985.
Cf. A358080, A370927.

Programs

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

Formula

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

A377548 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 - x*exp(x))^3 ).

Original entry on oeis.org

1, 3, 36, 789, 25644, 1112655, 60584058, 3975599271, 305587795320, 26941234079259, 2680537845979470, 297158198268036963, 36325021999771692036, 4854553774172042934279, 704185171457954845825026, 110192472149320674192100815, 18503193203651913813111781488, 3318723221891108953801703239731
Offset: 0

Views

Author

Seiichi Manyama, Oct 31 2024

Keywords

Links

Crossrefs

Cf. A213644, A377546.
Cf. A365177.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(serreverse(x*(1-x*exp(x))^3)/x))
  • PARI
    a(n) = 3*n!*sum(k=0, n, k^(n-k)*binomial(3*n+k+3, k)/((3*n+k+3)*(n-k)!));

Formula

E.g.f. satisfies A(x) = 1/(1 - x * A(x) * exp(x*A(x)))^3.
E.g.f.: B(x)^3, where B(x) is the e.g.f. of A365177.
a(n) = 3 * n! * Sum_{k=0..n} k^(n-k) * binomial(3*n+k+3,k)/( (3*n+k+3)*(n-k)! ).

A370930 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 - x*exp(x^2/2)) ).

Original entry on oeis.org

1, 1, 4, 33, 408, 6735, 139680, 3494715, 102486720, 3448812465, 131019940800, 5547190409145, 259025571826560, 13225167056035935, 733000949195074560, 43830500433645600675, 2812624056522882201600, 192798872614347464289825
Offset: 0

Views

Author

Seiichi Manyama, Mar 06 2024

Keywords

Links

Crossrefs

Cf. A213644, A370931.
Cf. A358264.

Programs

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

Formula

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

A370931 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 - x*exp(x^3/6)) ).

Original entry on oeis.org

1, 1, 4, 30, 340, 5180, 99360, 2300830, 62473600, 1946941920, 68507714800, 2686816932800, 116225776497600, 5497681373384200, 282305750023897600, 15640212734095950000, 929908726447266966400, 59061538103044360083200, 3990922849835432102592000
Offset: 0

Views

Author

Seiichi Manyama, Mar 06 2024

Keywords

Links

Crossrefs

Cf. A213644, A370930.
Cf. A358265.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(serreverse(x*(1-x*exp(x^3/6)))/x))
  • PARI
    a(n) = sum(k=0, n\3, (n-3*k)^k*(2*n-3*k)!/(6^k*k!*(n-3*k)!))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} (n-3*k)^k * (2*n-3*k)!/(6^k * k! * (n-3*k)!).

A377629 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 - x*exp(x))^4 ).

Original entry on oeis.org

1, 4, 60, 1644, 66712, 3611620, 245284344, 20071928212, 1923688610400, 211438912978692, 26225665058289640, 3624147718351890004, 552229557439437084816, 91990834731657653530180, 16632301623786709606057368, 3243982650658692575922907860, 678932992008068232965498759104
Offset: 0

Views

Author

Seiichi Manyama, Nov 02 2024

Keywords

Links

Crossrefs

Cf. A213644, A377546, A377548.
Cf. A377631, A377632.

Programs

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

Formula

E.g.f. satisfies A(x) = 1/(1 - x * A(x) * exp(x*A(x)))^4.
E.g.f.: B(x)^4, where B(x) is the e.g.f. of A377631.
a(n) = 4 * n! * Sum_{k=0..n} k^(n-k) * binomial(4*n+k+4,k)/( (4*n+k+4)*(n-k)! ).

A377631 E.g.f. satisfies A(x) = 1/(1 - x * A(x)^4 * exp(x*A(x)^4)).

Original entry on oeis.org

1, 1, 12, 297, 11380, 593785, 39304206, 3155996557, 298106913336, 32391139027185, 3980284376962330, 545806093612966021, 82628400115183659012, 13688201250584241332809, 2463065653446247669021398, 478399017659163635014545405, 99757368661138669886988396016
Offset: 0

Views

Author

Seiichi Manyama, Nov 02 2024

Keywords

Crossrefs

Cf. A377629, A377632.
Cf. A213644, A364985, A365177.
Cf. A364989.

Programs

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

Formula

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

A379684 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(x) * (1 - x*exp(x)) ).

Original entry on oeis.org

1, 0, 3, 11, 193, 2389, 50191, 1088205, 29836353, 902845673, 31428924631, 1207426391137, 51394833121105, 2386048646491197, 120379283952129567, 6547887322803355589, 382306453347573490177, 23839225109022069540817, 1581540933047988924532135
Offset: 0

Views

Author

Seiichi Manyama, Dec 29 2024

Keywords

Crossrefs

Cf. A213644, A377890, A379685.

Programs

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

Formula

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

A380095 E.g.f. A(x) satisfies A(x) = 1/sqrt( 1 - 2*x*A(x)^2*exp(x*A(x)^2) ).

Original entry on oeis.org

1, 1, 9, 156, 4129, 147880, 6696591, 367141306, 23648581713, 1750754472840, 146492770433095, 13672570280741086, 1408330043282040825, 158697952371711709060, 19420527592823261136519, 2564857285665551372127570, 363619232307437704055993761, 55079007956127598819416831088
Offset: 0

Views

Author

Seiichi Manyama, Jan 12 2025

Keywords

Crossrefs

Cf. A213644, A380096.
Cf. A380042, A379688.

Programs

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

Formula

E.g.f.: sqrt( (1/x) * Series_Reversion(x*(1 - 2*x*exp(x))) ).
a(n) = n! * Sum_{k=0..n} 2^k * k^(n-k) * binomial(n+k+1/2,k)/( (2*n+2*k+1)*(n-k)! ).
a(n) = (n!/(2*n+1)) * Sum_{k=0..n} (-2)^k * k^(n-k) * binomial(-n-1/2,k)/(n-k)!.
Previous Showing 11-20 of 29 results. Next

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