A113409 A transform of the central binomial coefficients A001405.
1, 1, 2, 3, 6, 11, 21, 39, 74, 141, 271, 521, 1004, 1939, 3756, 7291, 14176, 27599, 53805, 105031, 205268, 401573, 786328, 1541037, 3022528, 5932657, 11652617, 22901865, 45037432, 88616807, 174454943, 343606183, 677074350, 1334744305
Offset: 0
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
Programs
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Mathematica
Table[Sum[Binomial[n - k, k]*Binomial[k, Floor[k/2]], {k, 0, Floor[n/2]}], {n,0,50}] (* G. C. Greubel, Mar 09 2017 *) -
PARI
for(n=0,25, print1(sum(k=0,floor(n/2), binomial(n-k,k)*binomial(k,floor(k/2))), ", ")) \\ G. C. Greubel, Mar 09 2017
Formula
G.f.: (1-xc(x^2))/(1-x^2-x^4c(x^4)), where c(x) is the g.f. of A000108.
a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)*C(k, floor(k/2)).
a(n) = Sum_{k=0..n} C((n+k)/2, k)*C(floor((n-k)/2), floor((n-k)/4)).
Conjecture: (n+2)*a(n)-2*(n+1)*a(n-1) +(n-4)*a(n-2) +2*a(n-3) +4*(2-n)*a(n-4)=0. - R. J. Mathar, Nov 07 2012
a(n) ~ 2^(n + 3/2) / sqrt(3*Pi*n). - Vaclav Kotesovec, Nov 27 2017
Comments