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.

A190452 E.g.f. exp(x+x^2/2+x^4/24).

Original entry on oeis.org

1, 1, 2, 4, 11, 31, 106, 372, 1499, 6211, 28606, 135356, 697357, 3688049, 20935006, 121837276, 753159801, 4767863657, 31807384354, 217048147396, 1551200297291, 11327527814191, 86206555248122, 669666314150164, 5399592811359331, 44398500646885851
Offset: 0

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Author

Vladimir Kruchinin, May 24 2011

Keywords

Crossrefs

Column k=4 of A275422.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[Exp[x+x^2/2+x^4/24],{x,0,nn}], x]Range[0,nn]!] (* Harvey P. Dale, Jun 21 2012 *)
  • Maxima
    a(n):=n!*sum(sum(binomial(j,n-4*k+3*j)*12^(j-k)*binomial(k,j)*2^(-n+3*k-2*j),j,floor((4*k-n)/3),floor((4*k-n)/2))/k!,k,1,n);
    
  • PARI
    N=33;  x='x+O('x^N);
    egf=exp(x+x^2/2+x^4/4!);
    Vec(serlaplace(egf))
    /* Joerg Arndt, Sep 15 2012 */

Formula

E.g.f.: exp(x+x^2/2+x^4/24).
a(n) = n!*sum(k=1..n, sum(j=floor((4*k-n)/3)..floor((4*k-n)/2), binomial(j,n-4*k+3*j)*12^(j-k)*binomial(k,j)*2^(-n+3*k-2*j))/k!), n>0, a(0)=1.
Recurrence: 6*a(n) = 6*a(n-1) + 6*(n-1)*a(n-2) + (n-3)*(n-2)*(n-1)*a(n-4). - Vaclav Kotesovec, Oct 09 2013
a(n) ~ 1/2*exp((6*n)^(1/4) + sqrt(6*n)/2 - 3*n/4 - 3/4) * n^(3*n/4) * 6^(-n/4) * (1 + 3^(5/4)/(16*(2*n)^(3/4)) + 7*sqrt(3/2)/(8*sqrt(n)) - 3^(3/4)/(2*(2*n)^(1/4))). - Vaclav Kotesovec, Oct 09 2013

Extensions

More terms from Harvey P. Dale, Jun 21 2012