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%I #21 Apr 19 2024 11:30:14
%S 1,4,16,76,384,2004,10672,57628,314368,1728292,9560016,53144172,
%T 296642688,1661529588,9333781872,52566230076,296697618432,
%U 1677889961028,9505151782288,53928746972812,306393243712384,1742920028025364,9925790375394096,56584659163097436
%N Central terms of the triangle in A102413.
%C a(n) = A102413(2*n,n).
%H Reinhard Zumkeller, <a href="/A241023/b241023.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = 4 * A047781(n).
%F a(n) = 2*Hyper2F1([-n, n], [1], -1) for n>0. - _Peter Luschny_, Aug 02 2014
%F D-finite g.f. = (1+x)/sqrt(1-6*x+x^2), pairwise sums of A001850. - _R. J. Mathar_, Jan 15 2020
%F From _Peter Bala_, Apr 16 2024: (Start)
%F a(n) = Sum_{k = 0..n} (-1)^(n-k)*(2^k)*binomial(2*k, k)*binomial(n+k-1, n-k).
%F a(n) = (-1)^(n+1) * 4*n * hypergeom([n+1, -n+1], [2], 2).
%F n*(2*n - 3)*a(n) = 4*(3*n^2 - 6*n + 2)*a(n-1) - (2*n - 1)*(n - 2)*a(n-2) with a(0) = 1 and a(1) = 4.
%F O.g.f.: Sum_{n >= 0} (2^n)*binomial(2*n,n)*x^n/(1 + x)^(2*n) = 1 + 4*x + 16*x^2 + 76*x^3 + 384*x^4 + .... (End)
%t a[0] = 1; a[n_] := 4 Hypergeometric2F1[1 - n, n + 1, 1, -1];
%t Table[a[n], {n, 0, 23}] (* _Jean-François Alcover_, Jun 28 2019 *)
%o (Haskell)
%o a241023 n = a102413 (2 * n) n
%Y Cf. A001850, A047781, A102413.
%K nonn,easy
%O 0,2
%A _Reinhard Zumkeller_, Apr 15 2014
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