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

1, 3, 40, 875, 24696, 814968, 29899584, 1184303835, 49711519000, 2183727606632, 99503164453056, 4672502764108088, 225011739846443200, 11070183993903000000, 554749060302467136000, 28247778810831290434875, 1458696209123375067879000, 76266400563425844598365000

Offset

0,2

Links

Paolo Xausa, Table of n, a(n) for n = 0..500

Formula

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

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

a(n) = [x^n] hypergeom([1/2, -2*n - 1, -2*n], [2, 2], 4*x) (see A367023). - Peter Luschny, Nov 07 2023

Maple

C := n -> binomial(2*n, n)/(n + 1):
A358368 := n -> add(C(n)^2*binomial(n+k, k), k = 0..n): seq(A358368(n), n = 0..17);
# Alternative:
a := proc(n) option remember; if n = 0 then 1 else
(64*n^3 - 32*n^2 - 16*n + 8)*a(n - 1) / (n + 1)^3 fi end: seq(a(n), n = 0..17);
# Third form:
p := n -> hypergeom([1/2, -2*n - 1, -2*n], [2, 2], 4*x):
a := n -> coeff(simplify(p(n)), x, n): seq(a(n), n = 0..17);

Mathematica

Array[(2*#+1)*CatalanNumber[#]^3 &, 20, 0] (* Paolo Xausa, Feb 19 2024 *)

Crossrefs

Cf. A000108, A358436, A358437, A367023.

Sequence in context: A156356 A143640 A341849 * A260754 A047799 A204515

Adjacent sequences: A358365 A358366 A358367 * A358369 A358370 A358371

Keyword

nonn

Author

Peter Luschny, Nov 16 2022

Status

approved