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.

A377575 E.g.f. satisfies A(x) = (1 + x * exp(x) * A(x))^3.

Original entry on oeis.org

1, 3, 30, 483, 11100, 334035, 12478698, 558058179, 29104042152, 1735547479587, 116539815603630, 8704631976941043, 716019297815418732, 64326542671867079955, 6267631435921525638738, 658359915933162131600355, 74168964857766293453918928, 8921104769819780822122624323
Offset: 0

Views

Author

Seiichi Manyama, Nov 02 2024

Keywords

Crossrefs

Cf. A006153, A377574, A377576.
Cf. A364983, A377554.

Programs

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

Formula

E.g.f.: B(x)^3, where B(x) is the e.g.f. of A364983.
a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(3*k+3,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

A377577 E.g.f. satisfies A(x) = (1 + x * exp(x) * A(x)^2)^2.

Original entry on oeis.org

1, 2, 22, 426, 12344, 480010, 23500812, 1389576230, 96382531408, 7675512189714, 690344499939860, 69220070789605582, 7656687699685355256, 926243380308839330426, 121653259759077599227612, 17240419344948437264399670, 2622300119032920100004726432, 426102385668766701871015106338
Offset: 0

Views

Author

Seiichi Manyama, Nov 02 2024

Keywords

Crossrefs

Cf. A364987, A377576.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, k^(n-k)*binomial(4*k+2, k)/((2*k+1)*(n-k)!));

Formula

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A364987.
a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(4*k+2,k)/( (2*k+1)*(n-k)! ).

A377583 Expansion of e.g.f. (1 + x * exp(x))^4.

Original entry on oeis.org

1, 4, 20, 108, 616, 3620, 21624, 129892, 778208, 4621572, 27080680, 156080804, 883304976, 4905620356, 26743018904, 143219056740, 754280089024, 3911369843204, 19995029207496, 100885122939172, 502952669726960, 2480084192804484, 12107351426245240, 58565261434872548
Offset: 0

Views

Author

Seiichi Manyama, Nov 02 2024

Keywords

Crossrefs

Cf. A002999, A377582.
Cf. A377399, A377576.

Programs

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

Formula

a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(4,k)/(n-k)!.
G.f.: (1 - 36*x + 595*x^2 - 5970*x^3 + 40543*x^4 - 196752*x^5 + 702365*x^6 - 1871250*x^7 + 3740456*x^8 - 5614440*x^9 + 6362360*x^10 - 5588880*x^11 + 3979680*x^12 - 2196672*x^13 + 663552*x^14) / ((1-x)^2*(1-2*x)^3*(1-3*x)^4*(1-4*x)^5).
Showing 1-4 of 4 results.

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