A248152 a(n) = 48 * 4^n * (2*n-1)!! * (2*n+3)!! / ((n+2)! * (n+4)!).

3, 4, 14, 72, 462, 3432, 28314, 252824, 2401828, 23984688, 249554968, 2687515040, 29802622140, 338931781200, 3940081956450, 46695460462200, 562939717794300, 6890720756158800, 85507580292334200, 1074244300649863200, 13647785546892580200, 175166360584464768480

Offset: 0

Author

Michael Somos, Oct 02 2014

Keywords

• nonn

Comments

a(-1) = -2, a(-2) = 1/2 makes some formulas true for all n in Z.

Examples

			G.f. = 3 + 4*x + 14*x^2 + 72*x^3 + 462*x^4 + 3432*x^5 + 28314*x^6 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A000108, A005700.

Programs

Magma

[24*Factorial(2*n)*Factorial(2*n+3)/(Factorial(n)*Factorial(n+1)* Factorial(n+2)*Factorial(n+4)): n in [0..30]]; // G. C. Greubel, Aug 04 2018

Mathematica

a[ n_] := 48 4^n (2 n - 1)!! * (2 n + 3)!! / ((n + 2)! (n + 4)!);

PARI

{a(n) = if( n<0, if( n<-2, 0, [-2, 1/2][-n]), 24 * (2*n)! * (2*n+3)! / ( n! * (n+1)! * (n+2)! * (n+4)! ))};

Formula

a(n) = C(n)*C(n+3) - C(n+1)*C(n+2) for n>=0 where C() is the Catalan numbers A000108.

a(n)*a(n+1) = 4*A005700(n)*A005700(n+2) for all n in Z.

a(n) = 24 * (2*n)! * (2*n+3)! / ( n! * (n+1)! * (n+2)! * (n+4)! ) for n>=0.

(n+4)*(n+2)*a(n) -4*(2*n+3)*(2*n-1)*a(n-1)=0.

0 = a(n)*(+3840*a(n+3) - 660*a(n+4)) + a(n+1)*(+256*a(n+2) + 144*a(n+3) + 15*a(n+4)) + a(n+2)*(+4*a(n+2) + a(n+3)) for all n in Z.

0 = a(n)^2*(+196608*a(n+1)^2 - 14336*a(n+1)*a(n+2) - 25872*a(n+2)^2) + a(n)*a(n+1)*(+67584*a(n+1)^2 + 16672*a(n+1)*a(n+2) - 56*a(n+2)^2) + a(n+1)^2*(+1008*a(n+1)^2 + 264*a(n+1)*a(n+2) + 3*a(n+2)^2) for all n in Z.