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.

A088693 E.g.f: A(x) = f(x*A(x)^2), where f(x) = (1+3*x)*exp(x).

Original entry on oeis.org

1, 4, 71, 2434, 126117, 8804776, 775425427, 82565249670, 10319537275913, 1481520436347628, 240291243489544191, 43458295155840595306, 8672066947756086825325, 1892794863486905965709136, 448582856421716543783775947, 114720816495997657177701763246
Offset: 0

Views

Author

Paul D. Hanna, Oct 07 2003

Keywords

Comments

Radius of convergence of A(x): r = (2/27)*exp(-1/3) = 0.053076..., where A(r) = (3/2)*exp(1/6) and r = limit a(n)/a(n+1)*(n+1) as n->infinity. Radius of convergence is from a general formula yet unproved.

Crossrefs

Cf. A088690, A088692.

Programs

  • Mathematica
    Table[n!*SeriesCoefficient[((1+3*x)*E^x)^(2*n+1)/(2*n+1),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jan 24 2014 *)
  • PARI
    a(n)=n!*polcoeff(((1+3*x)*exp(x))^(2*n+1)+x*O(x^n),n,x)/(2*n+1)

Formula

a(n) = n! * [x^n] ((1+3*x)*exp(x))^(2*n+1)/(2*n+1).
a(n) ~ 3^(3*n+2) * n^(n-1) / (sqrt(7) * 2^(n+2) * exp(2*n/3-1/6)). - Vaclav Kotesovec, Jan 24 2014

A376123 E.g.f. A(x) satisfies A(x) = x * exp(A(x)) * (1 + 2*A(x)).

Original entry on oeis.org

0, 1, 6, 69, 1216, 29145, 886176, 32692597, 1419067392, 70867571409, 4002985561600, 252350116482981, 17564151708647424, 1337849793390444841, 110694246048458612736, 9886625352559043695125, 948044647019001482838016, 97146789899768662622795553
Offset: 0

Views

Author

Seiichi Manyama, Sep 11 2024

Keywords

Links

Crossrefs

Cf. A052885, A376124.
Cf. A088692, A376093, A376100.

Programs

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

Formula

E.g.f.: Series_Reversion( x * exp(-x) / (1 + 2*x) ).
a(n) = n! * Sum_{k=1..n} 2^(n-k) * n^(k-1) * binomial(n-1,k-1)/k!.
a(n) = n * A088692(n-1).
a(n) ~ 2^(2*n) * n^(n-1) / (sqrt(3) * exp(n/2)). - Vaclav Kotesovec, Sep 11 2024

A088694 E.g.f: A(x) = f(x*A(x)^3), where f(x) = (1+4*x)*exp(x).

Original entry on oeis.org

1, 5, 159, 10228, 1009253, 135069696, 22882888555, 4696799559488, 1133128780421385, 314294095403352064, 98550149514670698071, 34473870245560804316160, 13310522831484403851847981, 5622806397207798234900070400, 2579680348909056700728913816227
Offset: 0

Views

Author

Paul D. Hanna, Oct 07 2003

Keywords

Comments

Radius of convergence of A(x): r = (3^2/4^4)*exp(-1/4) = 0.0273797..., where A(r) = (4/3)*exp(1/12) and r = limit a(n)/a(n+1)*(n+1) as n->infinity. Radius of convergence is from a general formula yet unproved.

Crossrefs

Cf. A088690, A088692, A088693.

Programs

  • Mathematica
    Table[n!*SeriesCoefficient[((1+4*x)*E^x)^(3*n+1)/(3*n+1),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jan 24 2014 *)
  • PARI
    a(n)=n!*polcoeff(((1+4*x)*exp(x))^(3*n+1)+x*O(x^n),n,x)/(3*n+1)

Formula

a(n) = n! * [x^n] ((1+4*x)*exp(x))^(3*n+1)/(3*n+1).
a(n) ~ 16^(2*n+1) * n^(n-1) / (sqrt(13) * 9^(n+1) * exp(3*n/4 - 1/12)). - Vaclav Kotesovec, Jan 24 2014
Showing 1-3 of 3 results.

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