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A244039 a(n) = 2^(2*n-1) * ( binomial(3*n/2,n) + binomial((3*n-1)/2,n) ).

%I A244039 #39 Aug 08 2025 05:34:48
%S A244039 1,5,39,338,3075,28770,274134,2645844,25781283,253068530,2498678754,
%T A244039 24788450076,246889978062,2467197059124,24725226928140,
%U A244039 248396412496488,2500825206700323,25225687837101330,254877697946626410,2579123090162503500,26133512970919973850,265126176290618366460
%N A244039 a(n) = 2^(2*n-1) * ( binomial(3*n/2,n) + binomial((3*n-1)/2,n) ).
%D A244039 R. P. Stanley, Enumerative Combinatorics Volume 2, Cambridge Univ. Press, 1999, Theorem 6.33, p. 197.
%H A244039 Vincenzo Librandi, <a href="/A244039/b244039.txt">Table of n, a(n) for n = 0..200</a>
%H A244039 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_3(n).
%F A244039 From _Peter Bala_, Mar 04 2022: (Start)
%F A244039 a(n) = [x^n] ( (1 + 2*x)^3/(1 + x) )^n. Cf. A091527.
%F A244039 a(n) = Sum_{k = 0..n} (-1)^k * 2^(n-k) * binomial(3*n,n-k) * binomial(n+k-1,k).
%F A244039 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.
%F A244039 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.
%F A244039 Calculation gives (1 - 108*x^2)*A(x)^3 - (1 + 9*x)*A(x) = x.
%F A244039 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)
%F A244039 a(n) = 2^n*binomial(3*n, n)*hypergeom([-n, n], [2*n + 1], 1/2). - _Peter Luschny_, Mar 07 2022
%F A244039 From _Seiichi Manyama_, Aug 08 2025: (Start)
%F A244039 a(n) = Sum_{k=0..n} binomial(3*n,k) * binomial(2*n-k,n-k).
%F A244039 a(n) = [x^n] (1+x)^(3*n)/(1-x)^(n+1).
%F A244039 a(n) = [x^n] 1/((1-x)^n * (1-2*x)^(n+1)).
%F A244039 a(n) = Sum_{k=0..n} 2^k * binomial(n+k,k) * binomial(2*n-k-1,n-k). (End)
%p A244039 a := n -> 2^(2*n-1)*(binomial(3*n/2,n) + binomial((3*n-1)/2,n));
%p A244039 seq(a(n), n=0..25);
%t A244039 Table[2^(2n-1)*(Binomial[3n/2, n] + Binomial[(3n-1)/2, n]), {n, 0, 25}] (* _Vincenzo Librandi_, Jun 29 2014 *)
%o A244039 (PARI) a(n) = 2^(2*n-1)*(binomial(3*n/2, n) + binomial((3*n-1)/2, n));
%o A244039 vector(25, n, n--; a(n)) \\ _G. C. Greubel_, Aug 20 2019
%o A244039 (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
%o A244039 (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
%Y A244039 Cf. A045741, A091527, A244038, A244469.
%K A244039 nonn
%O A244039 0,2
%A A244039 _N. J. A. Sloane_, Jun 28 2014