A086905 a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(k,floor(k/2)).

1, 0, 2, 1, 5, 5, 15, 20, 50, 76, 176, 286, 638, 1078, 2354, 4081, 8789, 15521, 33099, 59279, 125477, 227239, 478193, 873885, 1830271, 3370029, 7030571, 13027729, 27088871, 50469889, 104647631, 195892564, 405187826, 761615284, 1571990936

Offset

0,3

Comments

Knödel walks starting and ending at 0, with n steps.

Links

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

H. Prodinger, The Kernel Method: a collection of examples, Séminaire Lotharingien de Combinatoire, B50f (2004), 19 pp.

Formula

G.f.: (sqrt((1+2*x)/(1-2*x))-1)/2/x/(1+x).

a(n) ~ 2^(n+3/2) / (3*sqrt(Pi*n)) * (1 - 2/(3*n)+ 3*(-1)^n/(4*n)). - Vaclav Kotesovec, Mar 02 2014

Mathematica

Table[Sum[(-1)^(n-k)*Binomial[k, Floor[k/2]], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 02 2014 *)

Prog

(PARI) a(n) = sum(k=0, n, (-1)^(n-k)*binomial(k, k\2)); \\ Michel Marcus, Dec 04 2016

Crossrefs

Cf. A036256, A001405.

First column of triangle A101491.

Sequence in context: A119245 A128731 A129157 * A167638 A317878 A209108

Adjacent sequences: A086902 A086903 A086904 * A086906 A086907 A086908

Keyword

nonn

Author

Vladeta Jovovic, Sep 19 2003

Status

approved