A113409 - OEIS

%I #10 Nov 27 2017 12:22:52

%S 1,1,2,3,6,11,21,39,74,141,271,521,1004,1939,3756,7291,14176,27599,

%T 53805,105031,205268,401573,786328,1541037,3022528,5932657,11652617,

%U 22901865,45037432,88616807,174454943,343606183,677074350,1334744305

%N A transform of the central binomial coefficients A001405.

%C Row sums of A113408.

%H G. C. Greubel, <a href="/A113409/b113409.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: (1-xc(x^2))/(1-x^2-x^4c(x^4)), where c(x) is the g.f. of A000108.

%F a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)*C(k, floor(k/2)).

%F a(n) = Sum_{k=0..n} C((n+k)/2, k)*C(floor((n-k)/2), floor((n-k)/4)).

%F 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

%F a(n) ~ 2^(n + 3/2) / sqrt(3*Pi*n). - _Vaclav Kotesovec_, Nov 27 2017

%t 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 *)

%o (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

%K easy,nonn

%O 0,3

%A _Paul Barry_, Oct 28 2005