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.

A382086 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x * C(x)) ), where C(x) = 1 + x*C(x)^2 is the g.f. of A000108.

Original entry on oeis.org

1, 1, 5, 52, 845, 18816, 533617, 18404800, 748039833, 35016198400, 1855389108221, 109781344134144, 7174844881882405, 513331696318615552, 39905830821183755625, 3349445733955326754816, 301886246619209909215793, 29080090017105458412257280, 2981488457660004727761477493
Offset: 0

Views

Author

Seiichi Manyama, Mar 15 2025

Keywords

Crossrefs

Cf. A382087, A382088, A382089.
Cf. A000108, A377831, A382036.

Programs

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

Formula

E.g.f. A(x) satisfies A(x) = exp(x*A(x) * C(x*A(x))).
a(n) = (n-1)! * Sum_{k=0..n-1} (n+1)^(n-k-1) * binomial(n+k-1,k)/(n-k-1)! for n > 0.
E.g.f.: exp( Series_Reversion( x * (1-x) * exp(-x) ) ).
a(n) ~ phi^(3*n - 3/2) * n^(n-1) / (5^(1/4) * exp((n - 1/phi)/phi)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Mar 15 2025

A382087 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x * B(x)^2) ), where B(x) = 1 + x*B(x)^3 is the g.f. of A001764.

Original entry on oeis.org

1, 1, 7, 106, 2525, 82536, 3436867, 174045376, 10385025849, 713599868800, 55498397386751, 4819444051348224, 462246012357060373, 48531686994029295616, 5536163290789601602875, 681824639839489261060096, 90168540044259473683829873, 12744019609725371553920876544
Offset: 0

Views

Author

Seiichi Manyama, Mar 15 2025

Keywords

Crossrefs

Cf. A382086, A382088, A382089.
Cf. A001764, A377832, A382037.

Programs

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

Formula

E.g.f. A(x) satisfies A(x) = exp(x*A(x) * B(x*A(x))^2).
a(n) = (n-1)! * Sum_{k=0..n-1} (n+1)^(n-k-1) * binomial(2*n+k-1,k)/(n-k-1)! for n > 0.
E.g.f.: exp( Series_Reversion( x * (1-x)^2 * exp(-x) ) ).

A382088 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x * B(x)^3) ), where B(x) = 1 + x*B(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 1, 9, 178, 5549, 237456, 12945037, 858203872, 67035559257, 6029839290880, 613862192499281, 69777500840918784, 8760124051527691141, 1203852634738613966848, 179746834136205848167125, 28975042890917781500747776, 5015346425440407318539964593, 927775677566572703009955053568
Offset: 0

Views

Author

Seiichi Manyama, Mar 15 2025

Keywords

Crossrefs

Cf. A382086, A382087, A382089.
Cf. A002293, A377833, A382038.

Programs

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

Formula

E.g.f. A(x) satisfies A(x) = exp(x*A(x) * B(x*A(x))^3).
a(n) = (n-1)! * Sum_{k=0..n-1} (n+1)^(n-k-1) * binomial(3*n+k-1,k)/(n-k-1)! for n > 0.
E.g.f.: exp( Series_Reversion( x * (1-x)^3 * exp(-x) ) ).
Showing 1-3 of 3 results.

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