A224734 G.f.: exp( Sum_{n>=1} binomial(2*n,n)^2 * x^n/n ).

1, 4, 26, 216, 2075, 21916, 247326, 2930216, 36028117, 456089076, 5910983050, 78100285784, 1048696065394, 14275198859304, 196610207633100, 2735542102308752, 38400942393884068, 543307627503591440, 7740605626606127512, 110970838624540461472, 1599834676405793089013

Offset: 0

Author

Paul D. Hanna, Apr 16 2013

Keywords

• nonn

Comments

The o.g.f. A(x) is the fourth power of the o.g.f. of A158266. - Peter Bala, Jun 04 2015

Examples

			G.f.: A(x) = 1 + 4*x + 26*x^2 + 216*x^3 + 2075*x^4 + 21916*x^5 + 247326*x^6 +...
where
log(A(x)) = 2^2*x + 6^2*x^2/2 + 20^2*x^3/3 + 70^2*x^4/4 + 252^2*x^5/5 + 924^2*x^6/6 + 3432^2*x^7/7 + 12870^2*x^8/8 +...+ A000984(n)^2*x^n/n +...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..800

Crossrefs

Cf. A224732, A224735, A224736, A002894, A000984, A158266.

Programs

Mathematica

CoefficientList[Series[Exp[4*x*HypergeometricPFQ[{1, 1, 3/2, 3/2}, {2, 2, 2}, 16*x]], {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 27 2025 *)

PARI

{a(n)=polcoeff(exp(sum(k=1,n,binomial(2*k,k)^2*x^k/k)+x*O(x^n)),n)}
for(n=0,20,print1(a(n),", "))

Formula

Logarithmic derivative yields A002894.

a(n) ~ c * 16^n / n^2, where c = 0.4942922... - Vaclav Kotesovec, Mar 27 2025