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A113409
A transform of the central binomial coefficients A001405.
2
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
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
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Oct 28 2005
STATUS
approved