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

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

Offset: 0

Author

Paul D. Hanna, Nov 19 2014

Keywords

• nonn

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

• Seiichi Manyama, Table of n, a(n) for n = 0..200

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.