A182890 - OEIS

%I #48 May 26 2023 20:41:53

%S 0,1,2,5,14,36,94,247,646,1691,4428,11592,30348,79453,208010,544577,

%T 1425722,3732588,9772042,25583539,66978574,175352183,459077976,

%U 1201881744,3146567256,8237820025,21566892818,56462858429,147821682470,387002188980,1013184884470

%N Number of (1,0)-steps of weight 1 at level 0 in all weighted lattice paths in L_n.

%C The members of L_n are paths of weight n that start at (0,0) and end on the horizontal axis and whose steps are of the following four kinds: a (1,0)-step with weight 1, a (1,0)-step with weight 2, a (1,1)-step with weight 2, and a (1,-1)-step with weight 1. The weight of a path is the sum of the weights of its steps.

%H M. Bona and A. Knopfmacher, <a href="http://dx.doi.org/10.1007/s00026-010-0060-7">On the probability that certain compositions have the same number of parts</a>, Ann. Comb., 14 (2010), 291-306.

%H E. Munarini and N. Zagaglia Salvi, <a href="http://dx.doi.org/10.1016/S0012-365X(02)00378-3">On the Rank Polynomial of the Lattice of Order Ideals of Fences and Crowns</a>, Discrete Mathematics 259 (2002), 163-177.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,2,-1).

%F G.f: x/((1+x+x^2)*(1-3*x+x^2)).

%F a(n) = Sum_{k>=0} k*A182888(n,k).

%F a(n) = (A000045(2n+2) - ((-1)^n)*A010892(n))/4. - _John M. Campbell_, Dec 30 2016

%F a(n) = Sum_{m=0..n} C(2*n-2*m,2*m+1)/2. - _Vladimir Kruchinin_, Jan 24 2022

%e a(3)=5. Indeed, denoting by h (resp. H) the (1,0)-step of weight 1 (resp. 2), and u=(1,1), d=(1,-1), the five paths of weight 3 are ud, du, hH, Hh, and hhh; they contain 0+0+1+1+3=5 h-steps at level 0.

%p G:=z/(1+z+z^2)/(1-3*z+z^2): Gser:=series(G,z=0,33): seq(coeff(Gser,z,n),n=0..30);

%t Table[Sum[Binomial[2n+2-2k,2k-1]/2, {k,0,n+1}], {n,0,30}]; (* _Rigoberto Florez_, Apr 10 2023 *)

%o (Maxima) a(n):=1/2*sum(binomial(2*n-2*m, 2*m+1), m, 0, (2*n-1)/4); /* _Vladimir Kruchinin_, Jan 24 2022 */

%Y Cf. A182888.

%K nonn,easy

%O 0,3

%A _Emeric Deutsch_, Dec 12 2010