A358368 - OEIS

%I #11 Feb 19 2024 04:36:24

%S 1,3,40,875,24696,814968,29899584,1184303835,49711519000,

%T 2183727606632,99503164453056,4672502764108088,225011739846443200,

%U 11070183993903000000,554749060302467136000,28247778810831290434875,1458696209123375067879000,76266400563425844598365000

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

%H Paolo Xausa, <a href="/A358368/b358368.txt">Table of n, a(n) for n = 0..500</a>

%F a(n) = (2*n + 1) * C(n)^3.

%F a(n) = (64*n^3 - 32*n^2 - 16*n + 8)*a(n - 1) / (n + 1)^3, for n >= 1.

%F a(n) = [x^n] hypergeom([1/2, -2*n - 1, -2*n], [2, 2], 4*x) (see A367023). - _Peter Luschny_, Nov 07 2023

%p C := n -> binomial(2*n, n)/(n + 1):

%p A358368 := n -> add(C(n)^2*binomial(n+k,k), k = 0..n): seq(A358368(n), n = 0..17);

%p # Alternative:

%p a := proc(n) option remember; if n = 0 then 1 else

%p (64*n^3 - 32*n^2 - 16*n + 8)*a(n - 1) / (n + 1)^3 fi end: seq(a(n), n = 0..17);

%p # Third form:

%p p := n -> hypergeom([1/2, -2*n - 1, -2*n], [2, 2], 4*x):

%p a := n -> coeff(simplify(p(n)), x, n): seq(a(n), n = 0..17);

%t Array[(2*#+1)*CatalanNumber[#]^3 &, 20, 0] (* _Paolo Xausa_, Feb 19 2024 *)

%Y Cf. A000108, A358436, A358437, A367023.

%K nonn

%O 0,2

%A _Peter Luschny_, Nov 16 2022