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.

A380723 E.g.f. A(x) satisfies A(x) = exp(x * A(x)^2) / (1 - x*A(x)^2).

Original entry on oeis.org

1, 2, 21, 436, 13785, 589206, 31825381, 2080523880, 159761186577, 14097898530730, 1405926737063541, 156379679761925148, 19195200442017128425, 2577494115099820986174, 375845854490491567916805, 59145488004443221188738256, 9990898494797767848442559649, 1803160967691789114062089511250
Offset: 0

Views

Author

Seiichi Manyama, Jan 30 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!.

A380727 E.g.f. A(x) satisfies A(x) = exp(x * A(x)^3 / (1 - x*A(x)^3)) / (1 - x*A(x)^3).

Original entry on oeis.org

1, 2, 31, 988, 48533, 3240016, 274099723, 28110919712, 3389978711785, 470124480093184, 73718009095023191, 12897488652935429632, 2490884805057416903869, 526368104133213244928000, 120811269372167469194820547, 29928528196949304888405323776, 7959458742917430589011715194833
Offset: 0

Views

Author

Seiichi Manyama, Jan 31 2025

Keywords

Crossrefs

Programs

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

Formula

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

A380946 Expansion of e.g.f. (1/x) * Series_Reversion( x * (1 - x)^3 * exp(-3*x) ).

Original entry on oeis.org

1, 6, 111, 3678, 179073, 11588688, 938905551, 91542271824, 10444685410881, 1365936450693120, 201503447217869679, 33108736185915906816, 5997057218957213126721, 1187319940110958086623232, 255104922613608981003351375, 59120580081196768991316314112
Offset: 0

Views

Author

Seiichi Manyama, Feb 09 2025

Keywords

Crossrefs

Programs

  • PARI
    a(n, q=3, r=3, s=3, 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(3*x*A(x))/(1 - x*A(x))^3.
E.g.f.: B(x)^3, where B(x) is the e.g.f. of A380724.
a(n) = 3 * n! * Sum_{k=0..n} (3*n+3)^(k-1) * binomial(4*n-k+2,n-k)/k!.
Showing 1-3 of 3 results.