%I #12 Nov 22 2024 07:24:00
%S 3,4,14,72,462,3432,28314,252824,2401828,23984688,249554968,
%T 2687515040,29802622140,338931781200,3940081956450,46695460462200,
%U 562939717794300,6890720756158800,85507580292334200,1074244300649863200,13647785546892580200,175166360584464768480
%N a(n) = 48 * 4^n * (2*n-1)!! * (2*n+3)!! / ((n+2)! * (n+4)!).
%C a(-1) = -2, a(-2) = 1/2 makes some formulas true for all n in Z.
%H G. C. Greubel, <a href="/A248152/b248152.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) = C(n)*C(n+3) - C(n+1)*C(n+2) for n>=0 where C() is the Catalan numbers A000108.
%F a(n)*a(n+1) = 4*A005700(n)*A005700(n+2) for all n in Z.
%F a(n) = 24 * (2*n)! * (2*n+3)! / ( n! * (n+1)! * (n+2)! * (n+4)! ) for n>=0.
%F (n+4)*(n+2)*a(n) -4*(2*n+3)*(2*n-1)*a(n-1)=0.
%F 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.
%F 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.
%e G.f. = 3 + 4*x + 14*x^2 + 72*x^3 + 462*x^4 + 3432*x^5 + 28314*x^6 + ...
%t a[ n_] := 48 4^n (2 n - 1)!! * (2 n + 3)!! / ((n + 2)! (n + 4)!);
%o (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)! ))};
%o (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
%Y Cf. A000108, A005700.
%K nonn,changed
%O 0,1
%A _Michael Somos_, Oct 02 2014