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 1-4 of 4 results.

A088695 E.g.f. satisfies A(x) = f(x*A(x)), where f(x) = exp(x+x^2).

Original entry on oeis.org

1, 1, 5, 40, 485, 7776, 156457, 3788800, 107414505, 3491200000, 128019454541, 5229222395904, 235490648957005, 11592449531084800, 619331166211640625, 35691050995648823296, 2206955604752999720273, 145757527499874820423680, 10240455593560436925898645
Offset: 0

Views

Author

Paul D. Hanna, Oct 07 2003

Keywords

Comments

Radius of convergence of A(x): r = (1/2)*exp(-3/4) = 0.23618..., where A(r) = exp(3/4) and r = limit a(n)/a(n+1)*(n+1) as n->infinity. Radius of convergence is from a general formula yet unproved.

Links

Crossrefs

Cf. A143768, A362771.
Cf. A365031, A365032.

Programs

  • Mathematica
    Table[n!*SeriesCoefficient[(E^(x+x^2))^(n+1)/(n+1),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jan 24 2014 *)
  • PARI
    a(n)=n!*polcoeff(exp(x+x^2)^(n+1)+x*O(x^n),n,x)/(n+1)

Formula

a(n) = n! * [x^n] exp(x+x^2)^(n+1)/(n+1).
a(n) = n! * Sum_{k=floor(n/2)..n} binomial(k,n-k)*(n+1)^(k-1)/k!. - Vladimir Kruchinin, Aug 04 2011
a(n) ~ 2^(n+1/2) * n^(n-1) / (sqrt(3) * exp(n/4 - 3/4)). - Vaclav Kotesovec, Jan 24 2014
E.g.f.: (1/x) * Series_Reversion( x*exp(-x*(1 + x)) ). - Seiichi Manyama, Sep 23 2024

A380781 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x * (1 + x)^2) / (1 + x)^2 ).

Original entry on oeis.org

1, 3, 29, 514, 13473, 470616, 20607781, 1086800352, 67105960641, 4750972007680, 379512594172941, 33771911612182272, 3313441417839023521, 355371388642280715264, 41365962922892138767125, 5193995331631149377867776, 699785874809076112607739009, 100701968551637581411176480768
Offset: 0

Views

Author

Seiichi Manyama, Feb 02 2025

Keywords

Links

Crossrefs

Cf. A377829, A380778, A380779, A380780.
Cf. A365031, A380762.
Cf. A380666.

Programs

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

Formula

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

A365032 E.g.f. satisfies A(x) = exp(x * A(x) * (1 + x * A(x))^3).

Original entry on oeis.org

1, 1, 9, 106, 1949, 47376, 1443757, 53003392, 2278044729, 112267072000, 6242682602321, 386708915902464, 26411820455554261, 1971959747016534016, 159794005364013403125, 13967707431203856449536, 1310083060716906045342833, 131245686122586065682628608
Offset: 0

Views

Author

Seiichi Manyama, Aug 17 2023

Keywords

Links

Crossrefs

Cf. A088695, A365031.
Cf. A364940.

Programs

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

Formula

a(n) = n! * Sum_{k=0..n} (n+1)^(k-1) * binomial(3*k,n-k)/k!.
E.g.f.: (1/x) * Series_Reversion( x*exp(-x*(1 + x)^3) ). - Seiichi Manyama, Sep 23 2024

A380762 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x * (1 + x)^2) / (1 + x) ).

Original entry on oeis.org

1, 2, 15, 208, 4249, 115656, 3946879, 162225680, 7807264497, 430828353280, 26825288214031, 1860715287986688, 142304071119852745, 11897080341213068288, 1079508321205459768575, 105660694801273960216576, 11097101798773200862180321, 1244852059489783737208012800
Offset: 0

Views

Author

Seiichi Manyama, Feb 02 2025

Keywords

Links

Crossrefs

Cf. A000129, A088690.
Cf. A365031, A380781.
Cf. A380664.

Programs

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

Formula

E.g.f. A(x) satisfies A(x) = exp( x * A(x) * (1 + x*A(x))^2 ) * (1 + x*A(x)).
a(n) = n! * Sum_{k=0..n} (n+1)^(k-1) * binomial(n+2*k+1,n-k)/k!.
Showing 1-4 of 4 results.

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