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A390104

G.f. A(x) satisfies A(x) = 1 + x/(1-x^3)^3 * A(x)^2.

1, 1, 2, 5, 17, 54, 177, 603, 2102, 7463, 26907, 98258, 362683, 1350990, 5072190, 19174068, 72919800, 278797240, 1071004920, 4131821244, 16001364200, 62184444840, 242426757780, 947841072984, 3715723010232, 14602017585789, 57512652626918, 226998536755430

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OFFSET

0,3

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

FORMULA

G.f.: 2/(1 + sqrt(1 - 4*x/(1-x^3)^3)).

a(n) = Sum_{k=0..floor(n/3)} binomial(3*n-8*k-1,k) * Catalan(n-3*k).

D-finite with recurrence: (n - 8)*a(n) + (11 - 4*n)*a(n + 3) + (15 + 6*n)*a(n + 6) + (16 + 4*n)*a(n + 8) + (-31 - 4*n)*a(n + 9) + (-46 - 4*n)*a(n + 11) + (13 + n)*a(n + 12) = 0. - Robert Israel, Feb 15 2026

MAPLE

f:= gfun:-rectoproc({(n - 8)*a(n) + (11 - 4*n)*a(n + 3) + (15 + 6*n)*a(n + 6) + (16 + 4*n)*a(n + 8) + (-31 - 4*n)*a(n + 9) + (-46 - 4*n)*a(n + 11) + (13 + n)*a(n + 12), a(0) = 1, a(1) = 1, a(2) = 2, a(3) = 5, a(4) = 17, a(5) = 54, a(6) = 177, a(7) = 603, a(8) = 2102, a(9) = 7463, a(10) = 26907, a(11) = 98258, a(12) = 362683}, a(n), remember):

map(f, [$0..40]); # Robert Israel, Feb 15 2026

MATHEMATICA

Table[Sum[ Binomial[3*n-8*k-1, k]*CatalanNumber[n-3*k], {k, 0, Floor[n/3]}], {n, 0, 30}] (* Vincenzo Librandi, Oct 30 2025 *)

PROG

(PARI) a(n) = sum(k=0, n\3, binomial(3*n-8*k-1, k)*binomial(2*(n-3*k), n-3*k)/(n-3*k+1));

(Magma) [&+[Catalan(n-3*k) * Binomial(3*n-8*k-1, k): k in [0..Floor(n/3)]] : n in [0..30] ]; // Vincenzo Librandi, Oct 30 2025

CROSSREFS

Cf. A376574, A390103.

Cf. A360100, A390102.

Cf. A000108, A389895.

Sequence in context: A133510 A191640 A389874 * A148408 A002692 A148409

Adjacent sequences: A390101 A390102 A390103 * A390105 A390106 A390107