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A330801 a(n) = A080247(2*n, n), the central values of the Big-Schröder triangle.

%I A330801 #19 May 03 2023 06:44:44
%S A330801 1,4,30,264,2490,24396,244790,2496528,25763058,268243860,2812481870,
%T A330801 29653804824,314097641130,3339741725404,35626286189670,
%U A330801 381098437754912,4086504567333858,43912100376527652,472743964145437310,5097853987059017000,55054474579787825562
%N A330801 a(n) = A080247(2*n, n), the central values of the Big-Schröder triangle.
%H A330801 G. C. Greubel, <a href="/A330801/b330801.txt">Table of n, a(n) for n = 0..950</a>
%F A330801 a(n) = ((n+1)/(2*n+1))*Sum_{j=0..n} binomial(2*n+1, n+j+1) * binomial(2*n+j, j).
%F A330801 a(n) = ((n+1)/(2*n+1))*binomial(2*n+1, n+1)*hypergeom([-n, 2*n + 1], [n + 2], -1).
%F A330801 D-finite with recurrence 2*n*(2*n+1)*(7*n-13)*a(n) - (382*n^3 -983*n^2 +533*n -40)*a(n-1) + (n-2)*(786*n^2 -3290*n +3315)*a(n-2) + (2*n-5)*(37*n-39)*(n-3)*a(n-3) = 0. - _R. J. Mathar_, Jul 27 2022
%F A330801 a(n) ~ phi^(5*n + 1) / (2 * 5^(1/4) * sqrt(Pi*n)), where phi = A001622 is the golden ratio. - _Vaclav Kotesovec_, May 03 2023
%p A330801 a := n -> ((n+1)/(2*n+1))*binomial(2*n+1, n+1)*hypergeom([-n, 2*n+1], [n+2], -1):
%p A330801 seq(simplify(a(n)), n=0..20);
%p A330801 # Alternative:
%p A330801 alias(C = binomial):
%p A330801 a := n -> ((n+1)/(2*n+1))*add(C(2*n+1, n+j+1)*C(2*n+j, j), j=0..n):
%p A330801 seq(a(n), n=0..20);
%t A330801 a[n_]:= (1/Sqrt[Pi]) 4^n (1 + n) Gamma[1/2 + n] Hypergeometric2F1Regularized[-n, 1 + 2 n, 2 + n, -1]; Table[a[n], {n, 0, 20}]
%o A330801 (Magma)
%o A330801 A330801:= func< n | ((n+1)/(2*n+1))*(&+[Binomial(2*n+1, n+j+1)*Binomial(2*n+j, j): j in [0..n]]) >;
%o A330801 [A330801(n): n in [0..40]]; // _G. C. Greubel_, May 03 2023
%o A330801 (SageMath)
%o A330801 def A330801(n) -> int:
%o A330801     s = sum( binomial(2 * n + 1, n + j + 1) * binomial(2 * n + j, j)
%o A330801         for j in range(n + 1) )
%o A330801     return (s * (n + 1)) // (2 * n + 1)
%o A330801 print([A330801(n) for n in range(41)])  # _G. C. Greubel_, May 03 2023
%Y A330801 Row sums of A330798.
%Y A330801 Cf. A080247.
%K A330801 nonn
%O A330801 0,2
%A A330801 _Peter Luschny_, Jan 02 2020