OFFSET
0,2
REFERENCES
R. P. Stanley, Enumerative Combinatorics Volume 2, Cambridge Univ. Press, 1999, Theorem 6.33, p. 197.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
I. M. Gessel, A short proof of the Deutsch-Sagan congruence for connected non crossing graphs, arXiv preprint arXiv:1403.7656 [math.CO], 2014. See f_3(n).
FORMULA
From Peter Bala, Mar 04 2022: (Start)
a(n) = [x^n] ( (1 + 2*x)^3/(1 + x) )^n. Cf. A091527.
a(n) = Sum_{k = 0..n} (-1)^k*2^(n-k)*binomial(3*n,n-k)*binomial(n+k-1,k).
n*(n-1)*(6*n-11)*a(n) = - 18*(n-1)*a(n-1) + 12*(3*n-4)*(3*n-5)*(6*n-5)*a(n-2) with a(0) = 1 and a(1) = 5.
The o.g.f. A(x) = 1 + 5*x + 39*x^2 + 338*x^3 + ... is the diagonal of the bivariate rational function 1/(1 - t*(1 + 2*x)^3/(1 + x)) and hence is an algebraic function over the field of rational functions Q(x) by Stanley 1999, Theorem 6.33, p. 197.
Calculation gives (1 - 108*x^2)*A(x)^3 - (1 + 9*x)*A(x) = x.
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and positive integers n and k. (End)
a(n) = 2^n*binomial(3*n, n)*hypergeom([-n, n], [2*n + 1], 1/2). - Peter Luschny, Mar 07 2022
MAPLE
a := n -> 2^(2*n-1)*(binomial(3*n/2, n) + binomial((3*n-1)/2, n));
seq(a(n), n=0..25);
MATHEMATICA
Table[2^(2n-1)*(Binomial[3n/2, n] + Binomial[(3n-1)/2, n]), {n, 0, 25}] (* Vincenzo Librandi, Jun 29 2014 *)
PROG
(PARI) a(n) = 2^(2*n-1)*(binomial(3*n/2, n) + binomial((3*n-1)/2, n));
vector(25, n, n--; a(n)) \\ G. C. Greubel, Aug 20 2019
(Magma) [Round(2^(2*n-1)*( Gamma(3*n/2+1)/Gamma(n/2+1) + Gamma((3*n+1)/2)/Gamma((n+1)/2) )/Factorial(n)): n in [0..25]]; // G. C. Greubel, Aug 20 2019
(Sage) [2^(2*n-1)*(binomial(3*n/2, n) + binomial((3*n-1)/2, n)) for n in (0..25)] # G. C. Greubel, Aug 20 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Jun 28 2014
STATUS
approved