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 1-6 of 6 results.

A364983 E.g.f. satisfies A(x) = 1 + x*exp(x)*A(x)^3.

Original entry on oeis.org

1, 1, 8, 111, 2332, 66125, 2368086, 102616759, 5222638856, 305436798009, 20186656927210, 1488021110087171, 121044207712073196, 10771321471267219525, 1040877104088653696606, 108549742436141933697135, 12151467262433697322437136, 1453367472748861203540942065
Offset: 0

Views

Author

Seiichi Manyama, Aug 15 2023

Keywords

Crossrefs

Cf. A364984, A364985, A364986.
Cf. A006153, A295238, A364987.
Cf. A001764, A307678.

Programs

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

Formula

a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(3*k+1,k)/( (3*k+1)*(n-k)! ) = n! * Sum_{k=0..n} k^(n-k) * A001764(k)/(n-k)!.
a(n) ~ sqrt(3) * sqrt(1 + LambertW(4/27)) * n^(n-1) / (2^(3/2) * exp(n) * LambertW(4/27)^n). - Vaclav Kotesovec, Nov 11 2024

A364985 E.g.f. satisfies A(x) = 1 + x*A(x)^3*exp(x*A(x)^2).

Original entry on oeis.org

1, 1, 8, 123, 2884, 91445, 3664926, 177796759, 10132646840, 663644108169, 49123993335130, 4055804550134051, 369544757016476196, 36834870020525413213, 3987179241476814768854, 465777171342934543710255, 58407238852473276959363056, 7825395596421876706944643985
Offset: 0

Views

Author

Seiichi Manyama, Aug 15 2023

Keywords

Crossrefs

Cf. A364983, A364984, A364986.
Cf. A003168.

Programs

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

Formula

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

A364989 E.g.f. satisfies A(x) = 1 + x*A(x)^4*exp(x*A(x)^4).

Original entry on oeis.org

1, 1, 10, 207, 6628, 288885, 15969606, 1070760523, 84448152328, 7660906993737, 785932068816010, 89973000854464431, 11370915080258640204, 1572520778920744136029, 236212754707591898128270, 38299196311415039667233715, 6666717272317556205911393296
Offset: 0

Views

Author

Seiichi Manyama, Aug 15 2023

Keywords

Crossrefs

Cf. A161633, A364982, A364986.

Programs

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

Formula

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

A364984 E.g.f. satisfies A(x) = 1 + x*A(x)^3*exp(x*A(x)).

Original entry on oeis.org

1, 1, 8, 117, 2596, 77705, 2936406, 134228059, 7204913528, 444331053873, 30963240318250, 2406301353714731, 206354828717754036, 19357367027097743449, 1971809610601104110942, 216754216326949771274715, 25575749384428387961718256, 3224227609551980271408565985
Offset: 0

Views

Author

Seiichi Manyama, Aug 15 2023

Keywords

Crossrefs

Cf. A364983, A364985, A364986.
Cf. A108447.

Programs

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

Formula

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

A377554 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + x*exp(x))^3 ).

Original entry on oeis.org

1, 3, 30, 537, 14124, 493695, 21601458, 1137294039, 70064934600, 4947238170747, 394022075650590, 34951812094581723, 3417754921150904172, 365287875167708973831, 42368411854713294141834, 5300422308901745571018735, 711465905597330333014408848, 101995745742232833085109746803
Offset: 0

Views

Author

Seiichi Manyama, Nov 01 2024

Keywords

Links

Crossrefs

Cf. A161633, A377553.
Cf. A364986.

Programs

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

Formula

E.g.f. satisfies A(x) = (1 + x * A(x) * exp(x*A(x)))^3.
E.g.f.: B(x)^3, where B(x) is the e.g.f. of A364986.
a(n) = (n!/(n+1)) * Sum_{k=0..n} k^(n-k) * binomial(3*n+3,k)/(n-k)!.

A364982 E.g.f. satisfies A(x) = 1 + x*A(x)^2*exp(x*A(x)^2).

Original entry on oeis.org

1, 1, 6, 69, 1204, 28345, 842406, 30282385, 1278159240, 61979238513, 3395850105610, 207490382754721, 13989267347891628, 1031687145559176457, 82618837044274734126, 7139807492658000170865, 662286433378726179463696, 65635135687587192429274849
Offset: 0

Views

Author

Seiichi Manyama, Aug 15 2023

Keywords

Crossrefs

Cf. A213644, A295238.
Cf. A161633, A364986, A364989.
Cf. A001764.

Programs

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

Formula

a(n) = (n!/(2*n+1)) * Sum_{k=0..n} k^(n-k) * binomial(2*n+1,k)/(n-k)!.
Showing 1-6 of 6 results.

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

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