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 1-4 of 4 results.

A377576 E.g.f. satisfies A(x) = (1 + x * exp(x) * A(x))^4.

Original entry on oeis.org

1, 4, 52, 1116, 34408, 1394340, 70298424, 4248802516, 299752943200, 24196951718532, 2200519882434280, 222683725755611604, 24824104612186789584, 3023063956714780554628, 399343825987950226379416, 56879649386095684434783060, 8689968793295620150120679104
Offset: 0

Views

Author

Seiichi Manyama, Nov 02 2024

Keywords

Crossrefs

Cf. A006153, A377574, A377575.
Cf. A364987, A377577.

Programs

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

Formula

E.g.f.: B(x)^4, where B(x) is the e.g.f. of A364987.
a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(4*k+4,k)/( (k+1)*(n-k)! ).

A377574 E.g.f. satisfies A(x) = (1 + x * exp(x) * A(x))^2.

Original entry on oeis.org

1, 2, 14, 150, 2264, 44370, 1073772, 30998954, 1041094448, 39909978594, 1720526113460, 82422717484602, 4345035540566184, 250012958308399442, 15594180423126432428, 1048169467357831893930, 75535629221800163853152, 5810132660615400890909634, 475146028302302130377698404
Offset: 0

Views

Author

Seiichi Manyama, Nov 02 2024

Keywords

Crossrefs

Cf. A006153, A377575, A377576.
Cf. A002999, A295238, A377553.

Programs

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

Formula

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A295238.
a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(2*k+2,k)/( (k+1)*(n-k)! ).
a(n) ~ 2^(5/2) * sqrt(1 + LambertW(1/4)) * n^(n-1) / (LambertW(1/4)^n * exp(n)). - Vaclav Kotesovec, Nov 02 2024

A377741 E.g.f. satisfies A(x) = exp(x) * (1 + x * A(x))^3.

Original entry on oeis.org

1, 4, 37, 583, 13225, 394681, 14659537, 652829857, 33937422001, 2018665692721, 135274646371561, 10087017309339433, 828563190097478425, 74348364577760978329, 7236649495742795579809, 759466703902106082652321, 85492204279344776678878945, 10275933748282019792253453025
Offset: 0

Views

Author

Seiichi Manyama, Nov 05 2024

Keywords

Crossrefs

Cf. A072597, A377740.
Cf. A364983, A377575.

Programs

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

Formula

a(n) = n! * Sum_{k=0..n} (k+1)^(n-k-1) * binomial(3*k+3,k)/(n-k)!.
a(n) = A364983(n+1)/(n+1).

A377582 Expansion of e.g.f. (1 + x * exp(x))^3.

Original entry on oeis.org

1, 3, 12, 51, 228, 1035, 4698, 21063, 92424, 395091, 1643790, 6664383, 26387100, 102286587, 389125506, 1455994935, 5368721808, 19541252259, 70312410774, 250408115823, 883617559140, 3092276105163, 10740749281482, 37053754521831, 127037475064728, 433073722098675
Offset: 0

Views

Author

Seiichi Manyama, Nov 02 2024

Keywords

Crossrefs

Cf. A002999, A377583.
Cf. A001815, A052791.
Cf. A377530, A377575.

Programs

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

Formula

a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(3,k)/(n-k)!.
G.f.: (1-17*x+127*x^2-542*x^3+1453*x^4-2543*x^5+2973*x^6-2478*x^7+1626*x^8-648*x^9) / ((1-x)^2*(1-2*x)^3*(1-3*x)^4).
Showing 1-4 of 4 results.

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

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