This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A244038 #75 Aug 07 2025 09:21:43
%S A244038 1,6,48,420,3840,36036,344064,3325608,32440320,318704100,3148873728,
%T A244038 31256180280,311452237824,3113596420200,31213674823680,
%U A244038 313672599360720,3158823892156416,31870058661517860,322076161553203200,3259691964853493400,33034843349204336640,335189468043077792760
%N A244038 a(n) = 4^n * binomial(3*n/2,n).
%H A244038 Vincenzo Librandi, <a href="/A244038/b244038.txt">Table of n, a(n) for n = 0..200</a>
%H A244038 I. M. Gessel, <a href="http://arxiv.org/abs/1403.7656">A short proof of the Deutsch-Sagan congruence for connected non crossing graphs</a>, arXiv preprint arXiv:1403.7656 [math.CO], 2014. See f_1(n).
%F A244038 a(n) = A045741(n+1) + A244039(n) [Gessel].
%F A244038 a(n) = [x^n] 1/sqrt(1 - 4*x)^(n+2). - _Ilya Gutkovskiy_, Oct 10 2017
%F A244038 G.f. A(x) satisfies: A(x)^3 * (1 - 108*x^2) = 3*A(x) - 2. - _Michael Somos_, Jan 27 2018
%F A244038 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
%F A244038 G.f.: 2/(1-2*sin(arcsin(216*x^2-1)/3)). - _Vladimir Kruchinin_, Oct 06 2022
%F A244038 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
%F A244038 From _Seiichi Manyama_, Aug 07 2025: (Start)
%F A244038 a(n) = Sum_{k=0..n} binomial(3*n+1,k) * binomial(2*n-k,n-k).
%F A244038 a(n) = [x^n] (1+x)^(3*n+1)/(1-x)^(n+1).
%F A244038 a(n) = [x^n] 1/((1-x) * (1-2*x))^(n+1).
%F A244038 a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(3*n+1,k) * binomial(2*n-k,n-k).
%F A244038 a(n) = Sum_{k=0..n} 2^k * binomial(n+k,k) * binomial(2*n-k,n-k). (End)
%p A244038 f1:=n->4^n*binomial(3*n/2,n); [seq(f1(n),n=0..40)];
%t A244038 Table[4^n Binomial[3 n/2, n], {n, 0, 40}] (* _Vincenzo Librandi_, Jun 29 2014 *)
%o A244038 (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 */
%o A244038 (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
%Y A244038 Cf. A045741, A244039.
%Y A244038 Cf. A160906, A383326.
%Y A244038 Cf. A000302, A098430, A386897.
%K A244038 nonn
%O A244038 0,2
%A A244038 _N. J. A. Sloane_, Jun 28 2014