A129366 a(n) = Sum_{k=0..floor(n/2)} A000108(n-k).
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
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
Programs
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Magma
[(&+[Catalan(n-j): j in [0..Floor(n/2)]]): n in [0..30]]; // G. C. Greubel, Jan 31 2024
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Mathematica
Table[Sum[CatalanNumber[k], {k, Floor[(n + 1)/2], n}], {n, 0, 30}] (* Wesley Ivan Hurt, Jun 18 2022 *) -
SageMath
[sum(catalan_number(n-j) for j in range(1+int(n//2))) for n in range(31)] # G. C. Greubel, Jan 31 2024
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
Comments