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 1-5 of 5 results.

A380726 E.g.f. A(x) satisfies A(x) = exp(x * A(x)^2 / (1 - x*A(x)^2)) / (1 - x*A(x)^2).

Original entry on oeis.org

1, 2, 23, 526, 18345, 865426, 51606511, 3725086590, 315869177777, 30781410753250, 3390102419068071, 416446509483046318, 56455962861401232025, 8372599773137199223794, 1348414830158700569758655, 234364024637335981658563486, 43725325359127416298442233569
Offset: 0

Views

Author

Seiichi Manyama, Jan 31 2025

Keywords

Crossrefs

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (2*n+1)^(k-1)*binomial(3*n, n-k)/k!);

Formula

a(n) = n! * Sum_{k=0..n} (2*n+1)^(k-1) * binomial(3*n,n-k)/k!.

A380724 E.g.f. A(x) satisfies A(x) = exp(x * A(x)^3) / (1 - x*A(x)^3).

Original entry on oeis.org

1, 2, 29, 862, 39461, 2454296, 193406953, 18475039808, 2075062993865, 268013104242688, 39139481641977461, 6377306725457207552, 1147019426037344539501, 225728971809041691392000, 48248339461852786811399489, 11131014193619108036340637696, 2756799306857952163745291500433
Offset: 0

Views

Author

Seiichi Manyama, Jan 30 2025

Keywords

Crossrefs

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (3*n+1)^(k-1)*binomial(4*n-k, n-k)/k!);

Formula

a(n) = n! * Sum_{k=0..n} (3*n+1)^(k-1) * binomial(4*n-k,n-k)/k!.

A380753 E.g.f. A(x) satisfies A(x) = exp(x * A(x)^2) / (1 - x * A(x)^2)^2.

Original entry on oeis.org

1, 3, 47, 1453, 68349, 4344751, 348936139, 33912469305, 3871084443641, 507765120717691, 75265926888996711, 12443096536067016997, 2270083842550815380725, 453042725968243823206887, 98183026886745981671902979, 22962952582930039784948279281, 5764815614414943166224203759601
Offset: 0

Views

Author

Seiichi Manyama, Feb 01 2025

Keywords

Crossrefs

Programs

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

Formula

a(n) = n! * Sum_{k=0..n} (2*n+1)^(k-1) * binomial(5*n-k+1,n-k)/k!.

A380755 E.g.f. A(x) satisfies A(x) = exp(x * A(x)^2 / (1 - x * A(x)^2)^2) / (1 - x * A(x)^2).

Original entry on oeis.org

1, 2, 25, 622, 23601, 1211306, 78585241, 6171434550, 569338685089, 60362321078674, 7232765564919321, 966640735654507838, 142570635491126076625, 23003561321179411452858, 4030628821337323603113241, 762175215630679850520288646, 154707566043362563540600474689
Offset: 0

Views

Author

Seiichi Manyama, Feb 01 2025

Keywords

Crossrefs

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (2*n+1)^(k-1)*binomial(3*n+k, n-k)/k!);

Formula

a(n) = n! * Sum_{k=0..n} (2*n+1)^(k-1) * binomial(3*n+k,n-k)/k!.

A380945 Expansion of e.g.f. (1/x) * Series_Reversion( x * (1 - x)^2 * exp(-2*x) ).

Original entry on oeis.org

1, 4, 50, 1124, 37192, 1637232, 90278176, 5992556320, 465599728512, 41470892979200, 4167168740195584, 466428111222196224, 57556315795242096640, 7763511917730857967616, 1136484206117494859980800, 179453678311835212416585728, 30404317385796994658988752896
Offset: 0

Views

Author

Seiichi Manyama, Feb 09 2025

Keywords

Crossrefs

Programs

  • PARI
    a(n, q=2, r=2, s=2, t=0, u=1) = q*n!*sum(k=0, n, (r*n+(s-r)*k+q)^(k-1)*binomial((r*u+1)*n+((s-r)*u+t-1)*k+q*u-1, n-k)/k!);

Formula

E.g.f. A(x) satisfies A(x) = exp(2*x*A(x))/(1 - x*A(x))^2.
E.g.f.: B(x)^2, where B(x) is the e.g.f. of A380723.
a(n) = 2 * n! * Sum_{k=0..n} (2*n+2)^(k-1) * binomial(3*n-k+1,n-k)/k!.
Showing 1-5 of 5 results.