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A244469 a(0) = 0, thereafter, a(n) = 2^(2*n-1)*( binomial((3*n-1)/2,n) - binomial(3*n/2, n)/3 ).

%I A244469 #31 Sep 08 2022 08:46:08
%S A244469 0,1,7,58,515,4746,44758,428772,4154403,40599130,399429602,3950996556,
%T A244469 39255152846,391466112324,3916110379020,39281346256008,
%U A244469 394942611929379,3978982062756090,40160256911157610,405995113593507900,4110284071450416090,41666530928566504620,422876855107176561780
%N A244469 a(0) = 0, thereafter, a(n) = 2^(2*n-1)*( binomial((3*n-1)/2,n) - binomial(3*n/2, n)/3 ).
%H A244469 Vincenzo Librandi, <a href="/A244469/b244469.txt">Table of n, a(n) for n = 0..200</a>
%H A244469 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_4(n).
%F A244469 G.f.: g'(x)/g(x)-1, g(x)=(2*sqrt(9*x+1)*sin(arcsin((54*x^2+27*x+2)/(2*(9*x+1)^(3/2)))/3))/3-1/3. - _Vladimir Kruchinin_, Apr 14 2019
%F A244469 From _Peter Bala_, Mar 05 2022: (Start)
%F A244469 a(n) = (1/n)*Sum_{k = 0..n} k*2^(n-k)*binomial(n+k-1,k)*binomial(2*n-k-1,n-k) for n >= 1.
%F A244469 a(n) = [x^n] G(x)^n = [x^n] 1/(1 - x*C(2*x))^n, where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108 and G(x) is the g.f. of A064062.
%F A244469 n*(n-1)*(6*n-7)*a(n) = - 18*(n-1)*a(n-1) + 12*(3*n-5)*(6*n-1)*(3*n-4)*a(n-2) with a(1) = 1 a(2) = 7.
%F A244469 exp(Sum_{n >= 1} a(n)*x^n/n) = 1 + x + 4*x^2 + 23*x^3 + 156*x^4 + ... is the g.f. of A007297.
%F A244469 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)
%p A244469 f4:=n->-(2^(2*n-1)/3)*binomial(3*n/2,n) + 2^(2*n-1)*binomial((3*n-1)/2,n);
%p A244469 [seq(f4(n),n=1..40)]; # then prepend f4(0)=0.
%t A244469 Join[{0}, Table[-(2^(2 n - 1)/3) Binomial[3 n/2, n] + 2^(2 n - 1) Binomial[(3 n - 1)/2, n], {n, 1, 30}]] (* _Vincenzo Librandi_, Jun 29 2014 *)
%o A244469 (PARI) {a(n) = if(n==0,0, 2^(2*n-1)*(binomial((3*n-1)/2, n) - binomial(3*n/2, n)/3) )}; \\ _G. C. Greubel_, Apr 17 2019
%o A244469 (Magma) [n eq 0 select 0 else Round(2^(2*n-1)*(Gamma((3*n+1)/2)/Gamma((n+1)/2) - Gamma((3*n+2)/2)/(3*Gamma((n+2)/2)))/Factorial(n)): n in [0..30]]; // _G. C. Greubel_, Apr 17 2019
%o A244469 (Sage)
%o A244469 def a(n):
%o A244469    if n==0: return 0
%o A244469    else: return 2^(2*n-1)*(binomial((3*n-1)/2, n) - binomial(3*n/2, n)/3)
%o A244469 [a(n) for n in (0..30)] # _G. C. Greubel_, Apr 17 2019
%Y A244469 Cf. A000108, A007297, A064062, A091527, A244039.
%K A244469 nonn
%O A244469 0,3
%A A244469 _N. J. A. Sloane_, Jun 28 2014