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A241023 Central terms of the triangle in A102413.

%I A241023 #29 Sep 19 2024 02:11:59
%S A241023 1,4,16,76,384,2004,10672,57628,314368,1728292,9560016,53144172,
%T A241023 296642688,1661529588,9333781872,52566230076,296697618432,
%U A241023 1677889961028,9505151782288,53928746972812,306393243712384,1742920028025364,9925790375394096,56584659163097436
%N A241023 Central terms of the triangle in A102413.
%H A241023 Reinhard Zumkeller, <a href="/A241023/b241023.txt">Table of n, a(n) for n = 0..1000</a>
%F A241023 a(n) = 4 * A047781(n).
%F A241023 a(n) = A102413(2*n,n).
%F A241023 a(n) = 2*Hyper2F1([-n, n], [1], -1) for n>0. - _Peter Luschny_, Aug 02 2014
%F A241023 D-finite g.f. = (1+x)/sqrt(1-6*x+x^2), pairwise sums of A001850. - _R. J. Mathar_, Jan 15 2020
%F A241023 From _Peter Bala_, Apr 16 2024: (Start)
%F A241023 a(n) = Sum_{k = 0..n} (-1)^(n-k)*(2^k)*binomial(2*k, k)*binomial(n+k-1, n-k).
%F A241023 a(n) = (-1)^(n+1) * 4*n * hypergeom([n+1, -n+1], [2], 2).
%F A241023 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 A241023 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)
%F A241023 From _Peter Bala_, Sep 18 2024: (Start)
%F A241023 a(n) = [x^n] 1/S(-x)^(2*n), where S(x) = (1 - x - sqrt(1 - 6*x + x^2))/(2*x) is the o.g.f. of the large Schröder numbers A006318. Cf. A333481.
%F A241023 The Gauss congruences hold: a(n*p^r) == a(n*p^(r-1)) (mod p^r) for all primes p and positive integers n and r.
%F A241023 Conjecture: the supercongruences a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) hold for all primes p >= 5 and positive integers n and r. (End)
%t A241023 a[0] = 1; a[n_] := 4 Hypergeometric2F1[1 - n, n + 1, 1, -1];
%t A241023 Table[a[n], {n, 0, 23}] (* _Jean-François Alcover_, Jun 28 2019 *)
%o A241023 (Haskell)
%o A241023 a241023 n = a102413 (2 * n) n
%Y A241023 Cf. A001850, A006318, A047781, A102413, A333481.
%K A241023 nonn,easy
%O A241023 0,2
%A A241023 _Reinhard Zumkeller_, Apr 15 2014