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 1-3 of 3 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

A381984 E.g.f. A(x) satisfies A(x) = exp(x) * B(x), where B(x) = 1 + x*B(x)^3 is the g.f. of A001764.

Original entry on oeis.org

1, 2, 9, 94, 1649, 40146, 1246057, 47004014, 2087644449, 106709890114, 6170322084041, 398219508589662, 28376096583546769, 2212797385807852754, 187441592012756668329, 17139223549605292448686, 1682551982313514625386817, 176505773149909540258262274, 19704960849698723062181296009
Offset: 0

Views

Author

Seiichi Manyama, Mar 11 2025

Keywords

Comments

For each positive integer k, the sequence obtained by reducing a(n) modulo k is a periodic sequence with period dividing k. For example, modulo 6 the sequence becomes [1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, ...] with period 6. Cf. A047974. - Peter Bala, Mar 13 2025

Crossrefs

Cf. A381985, A381986, A381987.
Cf. A001763, A001764.

Programs

  • Maple
    seq(simplify(hypergeom([-n, 1/3, 2/3], [3/2], -27/4)), n = 0..18); # Peter Bala, Mar 13 2025
  • Mathematica
    Table[HypergeometricPFQ[{-n, 1/3, 2/3}, {3/2}, -27/4], {n, 0, 20}] (* Vaclav Kotesovec, Mar 14 2025 *)
  • PARI
    a(n) = n!*sum(k=0, n, binomial(3*k+1, k)/((3*k+1)*(n-k)!));

Formula

a(n) = n! * Sum_{k=0..n} A001764(k)/(n-k)!.
From Peter Bala, Mar 13 2025: (Start)
a(n) = hypergeom([-n, 1/3, 2/3], [3/2], -27/4).
2*(2*n + 1)*a(n) = (27*n^2 - 19*n + 4)*a(n-1) - 2*(n - 1)*(27*n - 25)*a(n-2) + 27*(n - 1)*(n - 2)*a(n-3) with a(0) = 0, a(1) = 2 and a(2) = 9. (End)
a(n) ~ 3^(3*n + 1/2) * n^(n + 1/2) / (2^(2*n + 3/2) * exp(n - 4/27) * n^(3/2)). - Vaclav Kotesovec, Mar 14 2025

A381985 E.g.f. A(x) satisfies A(x) = exp(x) * B(x*A(x)), where B(x) = 1 + x*B(x)^3 is the g.f. of A001764.

Original entry on oeis.org

1, 2, 13, 217, 5937, 223641, 10725433, 625007993, 42883208609, 3386452550689, 302545287708201, 30170153462509545, 3322052185576104049, 400328811249634307249, 52406094009429908677049, 7405663486143907784247481, 1123601498350780798756198209, 182173718779147621454796872769
Offset: 0

Views

Author

Seiichi Manyama, Mar 11 2025

Keywords

Crossrefs

Cf. A381984, A381986.
Cf. A001764, A002293, A346646, A364987.

Programs

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

Formula

Let F(x) be the e.g.f. of A364987. F(x) = B(x*A(x)) = exp( 1/3 * Sum_{k>=1} binomial(3*k,k) * (x*A(x))^k/k ).
a(n) = n! * Sum_{k=0..n} (k+1)^(n-k) * A002293(k)/(n-k)!.
Showing 1-3 of 3 results.

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