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

0,3

Comments

Row sums of A113408.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

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

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 *)

Prog

(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

Crossrefs

Sequence in context: A316356 A049856 A302017 * A092684 A366107 A123915

Adjacent sequences: A113406 A113407 A113408 * A113410 A113411 A113412

Keyword

easy,nonn

Author

Paul Barry, Oct 28 2005

Status

approved