A067296 Fifth column of triangle A028364.

42, 90, 255, 785, 2529, 8398, 28511, 98462, 344720, 1220532, 4362438, 15718149, 57028063, 208164950, 763915215, 2816707650, 10429892340, 38768134380, 144600329250, 541036998090, 2030157909234, 7637910332556, 28805161727462, 108877152118780, 412384598880464, 1564961285214824

Offset

0,1

Links

Table of n, a(n) for n=0..25.

Formula

a(n) = A028364(n+4, 4) = Sum_{k=0..4} C(k)*C(n+4-k), with the Catalan numbers C(n) = A000108(n).

a(n) = ((193n^4+1727*n^3+5303*n^2+6457*n+2520)/(8*(2*n+1)*(2*n+3)*(2*n+5)*(2*n+7)))*C(n+4).

G.f.: (c4(x)*c(x)-(c4(x)-1)/x)/x^4, with c4(x) = Sum_{k=0..4} C(k)*x^k and c(x) is the g.f. for Catalan numbers A000108.

a(n) ~ 193 * 2^(2*n+1) / (n^(3/2) * sqrt(Pi)). - Amiram Eldar, Oct 09 2025

Mathematica

With[{c = CatalanNumber}, a[n_] := Sum[c[k] * c[n+4-k], {k, 0, 4}]; Array[a, 25, 0]] (* Amiram Eldar, Oct 09 2025 *)

Crossrefs

Cf. A000108, A028364, A067295 (fourth column).

Sequence in context: A300603 A301328 A370521 * A044180 A044561 A169680

Adjacent sequences: A067293 A067294 A067295 * A067297 A067298 A067299

Keyword

nonn,easy

Author

Wolfdieter Lang, Feb 05 2002

Status

approved