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.

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

A381989 E.g.f. A(x) satisfies A(x) = exp(x) * B(x*A(x)^2), where B(x) = 1 + x*B(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 2, 19, 514, 22621, 1369546, 105616639, 9901346554, 1093292035609, 138977379784882, 19990424969236171, 3209995501651871890, 569216406245186726965, 110476637766622355475898, 23294266811686640511534199, 5302371488162151660366545866, 1295920217231693678343467474353
Offset: 0

Views

Author

Seiichi Manyama, Mar 12 2025

Keywords

Crossrefs

Cf. A381987, A381988.
Cf. A002293, A002295, A382001.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (2*k+1)^(n-k)*binomial(6*k+1, k)/((6*k+1)*(n-k)!));

Formula

Let F(x) be the e.g.f. of A382001. F(x) = B(x*A(x)^2) = exp( 1/4 * Sum_{k>=1} binomial(4*k,k) * (x*A(x)^2)^k/k ).
a(n) = n! * Sum_{k=0..n} (2*k+1)^(n-k) * A002295(k)/(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 1-3 of 3 results.

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