A244038 a(n) = 4^n*binomial(3*n/2,n).

1, 6, 48, 420, 3840, 36036, 344064, 3325608, 32440320, 318704100, 3148873728, 31256180280, 311452237824, 3113596420200, 31213674823680, 313672599360720, 3158823892156416, 31870058661517860, 322076161553203200, 3259691964853493400, 33034843349204336640, 335189468043077792760

Offset

0,2

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_1(n).

Formula

a(n) = A045741(n+1) + A244039(n) [Gessel].

a(n) = [x^n] 1/sqrt(1 - 4*x)^(n+2). - Ilya Gutkovskiy, Oct 10 2017

G.f. A(x) satisfies: A(x)^3 * (1 - 108*x^2) = 3*A(x) - 2. - Michael Somos, Jan 27 2018

a(n) = [x^n] ( (1 + 4*x)^(3/2) )^n. It follows that 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. - Peter Bala, Mar 05 2022

G.f.: 2/(1-2*sin(arcsin(216*x^2-1)/3)). - Vladimir Kruchinin, Oct 06 2022

G.f.: ((3^(5/6)*i + 3^(1/3))*(-18*i*z + sqrt(-324*z^2 + 3))^(1/3) - (3^(5/6)*i - 3^(1/3))*(18*i*z + sqrt(-324*z^2 + 3))^(1/3))/(2*sqrt(-324*z^2 + 3)), where i = sqrt(-1) is the imaginary unit. - Karol A. Penson, Oct 24 2024

Maple

f1:=n->4^n*binomial(3*n/2, n); [seq(f1(n), n=0..40)];

Mathematica

Table[4^n Binomial[3 n/2, n], {n, 0, 40}] (* Vincenzo Librandi, Jun 29 2014 *)

Prog

(PARI) {a(n) = if( n<1, n==0, 3^n * polcoeff( serreverse( x / (x+1) / 2 * sqrt((x+3) / (x+1) / 3 + x * O(x^n))), n))}; /* Michael Somos, Jan 27 2018 */
(Magma) [Round(4^n*Gamma(3*n/2+1)/(Gamma(n+1)*Gamma(n/2+1))): n in [0..40]]; // G. C. Greubel, Aug 06 2018

Crossrefs

Cf. A045741, A244039.

Sequence in context: A366942 A258790 A345077 * A037184 A179075 A175916

Adjacent sequences: A244035 A244036 A244037 * A244039 A244040 A244041

Keyword

nonn

Author

N. J. A. Sloane, Jun 28 2014

Status

approved