A216584 a(n) = A002426(n)*A000984(n); product of central trinomial coefficients and central binomial coefficients.

1, 2, 18, 140, 1330, 12852, 130284, 1348776, 14247090, 152618180, 1654120468, 18096447096, 199536967084, 2214714164600, 24720932068200, 277289164574640, 3123590583844530, 35318969120870820, 400692715550057700, 4559427798654821400, 52020436064931914580

Offset: 0

Author

Paul D. Hanna, Sep 08 2012

Keywords

• nonn

Examples

			L.g.f.: L(x) = 2*x + 18*x^2/2 + 140*x^3/3 + 1330*x^4/4 + 12852*x^5/5 + 130284*x^6/6 + ...
where
exp(L(x)) = 1 + 2*x + 11*x^2 + 66*x^3 + 485*x^4 + 3842*x^5 + 32712*x^6 + ... + A216585(n)*x^n/n + ...
The central trinomial coefficients (A002426) begin:
[1, 1, 3, 7, 19, 51, 141, 393, 1107, 3139, 8953, 25653, 73789, ...];
The central binomial coefficients (A000984) begin:
[1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, 184756, 705432, ...].
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..925

Crossrefs

Cf. A216585, A151341, A168595, A002426, A000984.

Programs

Mathematica

Table[Binomial[2*n, n]*Sum[ Binomial[n, 2*k]*Binomial[2*k, k], {k, 0, Floor[n/2]}], {n,0,50}] (* G. C. Greubel, Feb 27 2017 *)

PARI

{a(n) = polcoeff((1+x+x^2)^n,n) * polcoeff((1+2*x+x^2)^n,n)}

PARI

{a(n)=binomial(2*n,n)*sum(k=0,n\2,binomial(n,2*k)*binomial(2*k,k))}
for(n=0,21,print1(a(n),", "))

Formula

a(n) = binomial(2*n, n) * Sum_{k=0..floor(n/2)} binomial(n, 2*k)*binomial(2*k, k).

Logarithmic derivative of A216585, after ignoring initial term a(0).

a(n) = [x^n*y^n] ( 1 + (x + y)^2 + (x + y)^4 )^n. - Peter Bala, Feb 17 2020

G.f.: hypergeom([1/2, 1/2],[1],16*x/(1+4*x))/sqrt(1+4*x). - Mark van Hoeij, May 13 2025