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A358368 a(n) = Sum_{k=0..n} C(n)^2 * binomial(n + k, k), where C(n) is the n-th Catalan number.

%I A358368 #11 Feb 19 2024 04:36:24
%S A358368 1,3,40,875,24696,814968,29899584,1184303835,49711519000,
%T A358368 2183727606632,99503164453056,4672502764108088,225011739846443200,
%U A358368 11070183993903000000,554749060302467136000,28247778810831290434875,1458696209123375067879000,76266400563425844598365000
%N A358368 a(n) = Sum_{k=0..n} C(n)^2 * binomial(n + k, k), where C(n) is the n-th Catalan number.
%H A358368 Paolo Xausa, <a href="/A358368/b358368.txt">Table of n, a(n) for n = 0..500</a>
%F A358368 a(n) = (2*n + 1) * C(n)^3.
%F A358368 a(n) = (64*n^3 - 32*n^2 - 16*n + 8)*a(n - 1) / (n + 1)^3, for n >= 1.
%F A358368 a(n) = [x^n] hypergeom([1/2, -2*n - 1, -2*n], [2, 2], 4*x) (see A367023). - _Peter Luschny_, Nov 07 2023
%p A358368 C := n -> binomial(2*n, n)/(n + 1):
%p A358368 A358368 := n -> add(C(n)^2*binomial(n+k,k), k = 0..n): seq(A358368(n), n = 0..17);
%p A358368 # Alternative:
%p A358368 a := proc(n) option remember; if n = 0 then 1 else
%p A358368 (64*n^3 - 32*n^2 - 16*n + 8)*a(n - 1) / (n + 1)^3 fi end: seq(a(n), n = 0..17);
%p A358368 # Third form:
%p A358368 p := n -> hypergeom([1/2, -2*n - 1, -2*n], [2, 2], 4*x):
%p A358368 a := n -> coeff(simplify(p(n)), x, n): seq(a(n), n = 0..17);
%t A358368 Array[(2*#+1)*CatalanNumber[#]^3 &, 20, 0] (* _Paolo Xausa_, Feb 19 2024 *)
%Y A358368 Cf. A000108, A358436, A358437, A367023.
%K A358368 nonn
%O A358368 0,2
%A A358368 _Peter Luschny_, Nov 16 2022