A358436
a(n) = Sum_{j=0..n} C(n)*C(n-j), where C(n) is the n-th Catalan number.
Original entry on oeis.org
1, 2, 8, 45, 322, 2730, 26004, 268554, 2940080, 33635316, 398300344, 4849845000, 60429982144, 767721774800, 9916427702880, 129937069996965, 1724052965464890, 23129299114182030, 313351935000465900, 4282621342230699930, 58994556159403576140, 818487022124443918740
Offset: 0
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C := n -> binomial(2*n, n)/(n + 1):
A358436 := n -> add(C(n)*C(n-j), j = 0..n):
seq(A358436(n), n = 0..21);
A358368
a(n) = Sum_{k=0..n} C(n)^2 * binomial(n + k, k), where C(n) is the n-th Catalan number.
Original entry on oeis.org
1, 3, 40, 875, 24696, 814968, 29899584, 1184303835, 49711519000, 2183727606632, 99503164453056, 4672502764108088, 225011739846443200, 11070183993903000000, 554749060302467136000, 28247778810831290434875, 1458696209123375067879000, 76266400563425844598365000
Offset: 0
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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);
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Array[(2*#+1)*CatalanNumber[#]^3 &, 20, 0] (* Paolo Xausa, Feb 19 2024 *)
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