A129366 a(n) = Sum_{k=0..floor(n/2)} C(n-k), where C(n) = A000108(n).

1, 1, 3, 7, 21, 61, 193, 617, 2047, 6895, 23691, 82435, 290447, 1033215, 3707655, 13402071, 48759741, 178403101, 656041801, 2423300129, 8987420549, 33453670773, 124936234413, 467995789277, 1757899936601

Offset

0,3

Comments

Partial sums of (A129367 prefixed by an initial 1).

Links

Table of n, a(n) for n=0..24.

Formula

G.f.: (1/(1-x))*(c(x)-x*c(x^2)), where c(x) is the g.f. of A000108(n).

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

a(n) = Sum_{k=floor((n+1)/2)..n} C(k), where C(n) = A000108(n).

Conjecture: +n*(12*n+35)*(n-1)*a(n) +(n-1)*(12*n^2-701*n+1236)*a(n-1) +2*(6*n^3-385*n^2+2285*n-3432)*a(n-2) +4*(-405*n^3+5313*n^2-19970*n+23175)*a(n-3) +8*(156*n^3-1724*n^2+5498*n-5175)*a(n-4) +16*(393*n^3-4981*n^2+20393*n-26820)*a(n-5) -32*(n-5)*(93*n-268)*(2*n-9)*a(n-6) = 0. - R. J. Mathar, Feb 05 2015

Mathematica

Table[Sum[CatalanNumber[k], {k, Floor[(n + 1)/2], n}], {n, 0, 30}] (* Wesley Ivan Hurt, Jun 18 2022 *)

Crossrefs

Cf. A000108, A129367.

Sequence in context: A005355 A182399 A025235 * A270049 A166358 A148670

Adjacent sequences: A129363 A129364 A129365 * A129367 A129368 A129369

Keyword

easy,nonn

Author

Paul Barry, Apr 11 2007

Status

approved