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.

A382000 E.g.f. A(x) satisfies A(x) = 1 + x*exp(2*x)*A(x)^5.

Original entry on oeis.org

1, 1, 14, 342, 12872, 659280, 42828912, 3375009568, 312860626304, 33361836534144, 4023352486200320, 541461682626399744, 80448618080927609856, 13079749459734097573888, 2309915877337042992324608, 440332184936376095626076160, 90117169223076699520606896128
Offset: 0

Views

Author

Seiichi Manyama, Mar 12 2025

Keywords

Crossrefs

Cf. A336950, A381997, A381998, A381999, A382001.
Cf. A002294, A377526, A381986.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (2*k)^(n-k)*binomial(5*k+1, k)/((5*k+1)*(n-k)!));

Formula

a(n) = n! * Sum_{k=0..n} (2*k)^(n-k) * A002294(k)/(n-k)!.
a(n) ~ 2^(n-3) * n^(n-1) * sqrt(5*(1 + LambertW(512/3125))) / (exp(n) * LambertW(512/3125)^n). - Vaclav Kotesovec, Mar 22 2025

A377528 E.g.f. satisfies A(x) = 1/(1 - x * exp(x) * A(x))^4.

Original entry on oeis.org

1, 4, 60, 1548, 58456, 2930020, 183763704, 13866109012, 1224251041248, 123885272536452, 14140672597851880, 1797709847594145364, 251941291752251706576, 38593132701417704324356, 6415647343472197357272984, 1150373241484390263973203540, 221318733487356013660505462464
Offset: 0

Views

Author

Seiichi Manyama, Oct 30 2024

Keywords

Crossrefs

Cf. A295238, A377503, A377504.
Cf. A377526, A377527.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, k^(n-k)*binomial(5*k+3, k)/((k+1)*(n-k)!));

Formula

E.g.f.: B(x)^4, where B(x) is the e.g.f. of A377526.
a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(5*k+3,k)/( (k+1)*(n-k)! ).

A377527 E.g.f. satisfies A(x) = 1/(1 - x * exp(x) * A(x)^2)^2.

Original entry on oeis.org

1, 2, 26, 618, 22256, 1081770, 66401532, 4931389358, 430108545680, 43104305664594, 4881518010253460, 616559703960596022, 85935621525038617752, 13102417265843584412474, 2169337115977056447577820, 387609934848899388554651550, 74340899731294447790784890912
Offset: 0

Views

Author

Seiichi Manyama, Oct 30 2024

Keywords

Crossrefs

Cf. A377526, A377528.
Cf. A377503, A377529.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, k^(n-k)*binomial(5*k+1, k)/((2*k+1)*(n-k)!));

Formula

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A377526.
a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(5*k+1,k)/( (2*k+1)*(n-k)! ).

A381988 E.g.f. A(x) satisfies A(x) = exp(x) * B(x*A(x)), where B(x) = 1 + x*B(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 2, 15, 313, 10773, 510981, 30876463, 2267990159, 196204786025, 19539828320905, 2201822913234771, 276969947671828995, 38473403439454795837, 5849221857618942870029, 966078641687956464576119, 172251173569831561500070711, 32975613823747758363130520529, 6746227557293225645352382744593
Offset: 0

Views

Author

Seiichi Manyama, Mar 12 2025

Keywords

Crossrefs

Cf. A381987, A381989.
Cf. A002293, A002294, A346647, A377526.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (k+1)^(n-k)*binomial(5*k+1, k)/((5*k+1)*(n-k)!));

Formula

Let F(x) be the e.g.f. of A377526. F(x) = B(x*A(x)) = exp( 1/4 * Sum_{k>=1} binomial(4*k,k) * (x*A(x))^k/k ).
a(n) = n! * Sum_{k=0..n} (k+1)^(n-k) * A002294(k)/(n-k)!.
Showing 1-4 of 4 results.

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