OFFSET
0,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..925
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
EXAMPLE
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, ...].
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 *)
PROG
(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), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 08 2012
STATUS
approved