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A244038 a(n) = 4^n*binomial(3*n/2,n). 7
1, 6, 48, 420, 3840, 36036, 344064, 3325608, 32440320, 318704100, 3148873728, 31256180280, 311452237824, 3113596420200, 31213674823680, 313672599360720, 3158823892156416, 31870058661517860, 322076161553203200, 3259691964853493400, 33034843349204336640, 335189468043077792760 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
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
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
Sequence in context: A366942 A258790 A345077 * A037184 A179075 A175916
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Jun 28 2014
STATUS
approved

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)