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.

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A249939 E.g.f.: 1/(5 - 4*cosh(x)).

Original entry on oeis.org

1, 4, 100, 6244, 727780, 136330084, 37455423460, 14188457293924, 7087539575975140, 4514046217675793764, 3570250394992512270820, 3433125893070920512725604, 3944372161432193963534198500, 5336301013125557989981503385444, 8396749419933421378024498580446180
Offset: 0

Views

Author

Paul D. Hanna, Nov 19 2014

Keywords

Comments

a(n) = 4*A242858(2*n) for n>0.
a(n) = A249940(n)/3.
a(n) == 4 (mod 96) for n>0.

Examples

			E.g.f.: E(x) = 1 + 4*x^2/2! + 100*x^4/4! + 6244*x^6/6! + 727780*x^8/8! +...
where E(x) = 1/(5 - 4*cosh(x)) = -exp(x) / (2 - 5*exp(x) + 2*exp(2*x)).
ALTERNATE GENERATING FUNCTION.
E.g.f.: A(x) = 1 + 4*x + 100*x^2/2! + 6244*x^3/3! + 727780*x^4/4! +...
where 3*A(x) = 1 + 2*exp(x)/2 + 2*exp(4*x)/2^2 + 2*exp(9*x)/2^3 + 2*exp(16*x)/2^4 + 2*exp(25*x)/2^5 + 2*exp(36*x)/2^6 + 2*exp(49*x)/2^7 +...

Links

Crossrefs

Cf. A249938, A249940, A247082, A250914, A250915, A242858.
Cf. A210676, A210657, A028296, A094088, A210672, A210674.

Programs

  • PARI
    /* E.g.f.: 1/(5 - 4*cosh(x)) */
{a(n) = local(X=x+O(x^(2*n+1))); (2*n)!*polcoeff( 1/(5 - 4*cosh(X)), 2*n)}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    /* Formula for a(n): */
{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n) = if(n==0, 1, sum(k=1, (2*n+1)\3, 2*(3*k-1)! * Stirling2(2*n+1, 3*k)))}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    /* Formula for a(n): */
{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n) = if(n==0, 1, (4/3)*sum(k=0, 2*n, k! * Stirling2(2*n, k) ))}
for(n=0, 20, print1(a(n), ", "))

Formula

E.g.f.: 1/3 + (2/3)*Sum_{n>=1} exp(n^2*x) / 2^n = Sum_{n>=0} a(n)*x^n/n!.
a(n) = (4/3) * Sum_{k=0..2*n} k! * Stirling2(2*n, k) for n>0 with a(0)=1.
a(n) = Sum_{k=1..[(2*n+1)/3]} 2 * (3*k-1)! * Stirling2(2*n+1, 3*k) for n>0 with a(0)=3, after Vladimir Kruchinin in A242858.

A249940 E.g.f.: 1 + Sum_{n>=1} 2*exp(n^2*x) / 2^n.

Original entry on oeis.org

3, 12, 300, 18732, 2183340, 408990252, 112366270380, 42565371881772, 21262618727925420, 13542138653027381292, 10710751184977536812460, 10299377679212761538176812, 11833116484296581890602595500, 16008903039376673969944510156332, 25190248259800264134073495741338540
Offset: 0

Views

Author

Paul D. Hanna, Nov 19 2014

Keywords

Examples

			E.g.f.: A(x) = 3 + 12*x + 300*x^2/2! + 18732*x^3/3! + 2183340*x^4/4! +...
where the e.g.f. equals the infinite series:
A(x) = 1 + 2*exp(x)/2 + 2*exp(4*x)/2^2 + 2*exp(9*x)/2^3 + 2*exp(16*x)/2^4 + 2*exp(25*x)/2^5 + 2*exp(36*x)/2^6 +...
We also have the following series expansion:
3/(5 - 4*cosh(x)) = 3 + 12*x^2/2! + 300*x^4/4! + 18732*x^6/6! + 2183340*x^8/8! + 408990252*x^10/10! +...

Crossrefs

Cf. A242858, A249941.

Programs

  • Mathematica
    nmax=20; Table[(CoefficientList[Series[3/(5-4*Cosh[x]), {x, 0, 2*nmax}], x] * Range[0, 2*nmax]!)[[n]],{n,1,2*nmax+2,2}] (* Vaclav Kotesovec, Nov 29 2014 *)
  • PARI
    /* E.g.f.: 3/(5 - 4*cosh(x)): */
{a(n) = local(X=x+O(x^(2*n+1))); (2*n)!*polcoeff( 3/(5 - 4*cosh(X)), 2*n)}
for(n=0,20,print1(a(n),", "))
  • PARI
    /* E.g.f.: 1 + Sum_{n>=1} 2*exp(n^2*x)/2^n */
\p100 \\ set precision
{a(n) = round( n!*polcoeff(1+2*sum(m=1,500,exp(m^2*x +x*O(x^n))/2^m*1.), n))}
for(n=0,20,print1(a(n),", "))
  • PARI
    /* Formula for a(n): */
{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n) = if(n==0,3, sum(k=0, 2*n, 4*k! * Stirling2(2*n, k) ))}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    /* Formula for a(n): */
{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n) = if(n==0,3, 2*sum(k=1,(2*n+1)\3, (3*k)! * Stirling2(2*n+1,3*k) / k))}
for(n=0,20,print1(a(n),", "))

Formula

E.g.f.: 3/(5 - 4*cosh(x)) = Sum_{n>=0} a(n)*x^(2*n)/(2*n)!.
a(n) = Sum_{k=0..2*n} 4*k! * Stirling2(2*n, k) for n>0 with a(0)=3.
a(n) = Sum_{k=1..[(2*n+1)/3]} 2*(3*k)! * Stirling2(2*n+1,3*k) / k for n>0 with a(0)=3, after Vladimir Kruchinin in A242858.
a(n) ~ 2 * (2*n)! / (log(2))^(2*n+1). - Vaclav Kotesovec, Nov 29 2014
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