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.

A381986 E.g.f. A(x) satisfies A(x) = exp(x) * B(x*A(x)^2), where B(x) = 1 + x*B(x)^3 is the g.f. of A001764.

Original entry on oeis.org

1, 2, 17, 388, 14329, 727206, 46984729, 3689119624, 341097752657, 36302764864330, 4371463743828481, 587606216836328460, 87219196719691250185, 14168990447072685567214, 2500554381188629649979593, 476391652257266128440376336, 97447147561230881896398507553
Offset: 0

Views

Author

Seiichi Manyama, Mar 11 2025

Keywords

Crossrefs

Cf. A381984, A381985.
Cf. A001764, A002294, A382000.

Programs

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

Formula

Let F(x) be the e.g.f. of A382000. F(x) = B(x*A(x)^2) = exp( 1/3 * Sum_{k>=1} binomial(3*k,k) * (x*A(x)^2)^k/k ).
a(n) = n! * Sum_{k=0..n} (2*k+1)^(n-k) * A002294(k)/(n-k)!.

A381987 E.g.f. A(x) satisfies A(x) = exp(x) * B(x), where B(x) = 1 + x*B(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 2, 11, 160, 3941, 134486, 5851327, 309520436, 19283504585, 1382980764106, 112223497464371, 10165461405056552, 1016801830348902061, 111312715288354681310, 13237965546409421546471, 1699516550894276788156156, 234263144339070269872076177, 34507561203827621878485498386
Offset: 0

Views

Author

Seiichi Manyama, Mar 12 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 5 the sequence becomes [1, 2, 1, 0, 1, 1, 2, 1, 0, 1, ...] with period 5. In particular, a(5*n+3) == 0 (mod 5). Cf. A047974. - Peter Bala, Mar 13 2025

Crossrefs

Cf. A381984, A381988, A381989.
Cf. A002293, A365340.

Programs

  • Maple
    seq(simplify(hypergeom([-n, 1/2, 1/4, 3/4], [2/3, 4/3], -256/27)), n = 0..17); # Peter Bala, Mar 13 2025
  • Mathematica
    Table[HypergeometricPFQ[{-n, 1/2, 1/4, 3/4}, {2/3, 4/3}, -256/27], {n, 0, 20}] (* Vaclav Kotesovec, Mar 14 2025 *)
  • PARI
    a(n) = n!*sum(k=0, n, binomial(4*k+1, k)/((4*k+1)*(n-k)!));

Formula

a(n) = n! * Sum_{k=0..n} A002293(k)/(n-k)!.
From Peter Bala, Mar 13 2025: (Start)
a(n) = hypergeom([-n, 1/2, 1/4, 3/4], [2/3, 4/3], -256/27).
3*(3*n - 1)*(3*n + 1)*a(n) = n*(256*n^2 - 303*n + 95)*a(n-1) - 3*(n - 1)*(256*n^2 - 485*n + 245)*a(n-2) + 3*(256*n - 375)*(n - 1)*(n - 2)*a(n-3) - 256*(n - 1)*(n - 2)*(n - 3)*a(n-4) with a(0) = 1, a(1) = 2, a(2) = 11 and a(3) = 160. (End)
a(n) ~ 2^(8*n+1) * n^(n-1) / (3^(3*n + 3/2) * exp(n - 27/256)). - 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.

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

Content is available under The OEIS End-User License Agreement.