A182886 -id:A182886 - cp's OEIS frontend

A182887 Number of (1,0)-steps in all weighted lattice paths in L_n.

0, 1, 3, 7, 21, 60, 166, 463, 1281, 3521, 9645, 26322, 71606, 194283, 525897, 1420595, 3830445, 10311510, 27718028, 74410105, 199519155, 534400491, 1429944603, 3822761742, 10211093226, 27254110405, 72691102131, 193750155673, 516100470051

Offset: 0

Emeric Deutsch, Dec 11 2010

nonn

These are paths 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; a (1,-1)-step with weight 1. The weight of a path is the sum of the weights of its steps.

			a(3)=7. Indeed, denoting by h (H) the (1,0)-step of weight 1 (2), and u=(1,1), d=(1,-1), the five paths of weight 3 are ud, du, hH, Hh, and hhh; the total number of (1,0) steps in them are 0+0+2+2+3=7.

G. C. Greubel, Table of n, a(n) for n = 0..1000
M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306.
E. Munarini, N. Zagaglia Salvi, On the Rank Polynomial of the Lattice of Order Ideals of Fences and Crowns, Discrete Mathematics 259 (2002), 163-177.

Cf. A182884, A182886.

G:=z*(1+z)*(1-z-z^2)/((1-3*z+z^2)*(1+z+z^2))^(3/2): Gser:=series(G,z=0,32): seq(coeff(Gser,z,n),n=0..28);

CoefficientList[Series[x*(1+x)*(1-x-x^2)/((1-3*x+x^2)*(1+x+x^2))^(3/2), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 06 2016 *)

z='z+O('z^50); concat([0], Vec(z*(1+z)*(1-z-z^2)/((1-3*z+z^2)*(1+z+z^2))^(3/2))) \\ G. C. Greubel, Mar 26 2017

a(n) = A182884(n) + A182884(n-1).
a(n) = Sum_{k>=0} k*A182886(n,k).
G.f.: z*(1+z)*(1-z-z^2)/((1-3*z+z^2)*(1+z+z^2))^(3/2).
a(n) ~ ((3 + sqrt(5))/2)^n * sqrt(n) / (2*5^(1/4)*sqrt(Pi)). - Vaclav Kotesovec, Mar 06 2016
Conjecture: (n-1)*(182*n-279)*a(n) + (-230*n^2+11*n+643)*a(n-1) + (-450*n^2+1603*n-315)*a(n-2) + (-498*n^2+971*n+57)*a(n-3) + (-86*n^2+959*n-529)*a(n-4) + (134*n-59)*(n-3)*a(n-5) = 0. - R. J. Mathar, Jun 14 2016

A182885 Triangle read by rows: T(n,k) is the number of weighted lattice paths in L_n having k (1,0)-steps of weight 2. These are paths that start at (0,0) , end on the horizontal axis and whose steps are of the following four kinds: an (1,0)-step with weight 1; an (1,0)-step with weight 2; a (1,1)-step with weight 2; a (1,-1)-step with weight 1. The weight of a path is the sum of the weights of its steps.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 7, 3, 1, 13, 10, 3, 27, 29, 6, 1, 61, 66, 22, 4, 133, 157, 75, 10, 1, 287, 398, 201, 40, 5, 633, 975, 538, 155, 15, 1, 1407, 2334, 1506, 476, 65, 6, 3121, 5631, 4077, 1414, 280, 21, 1, 6943, 13602, 10695, 4320, 966, 98, 7, 15517, 32623, 27966, 12765, 3150, 462, 28, 1

Offset: 0

Emeric Deutsch, Dec 11 2010

Sum of entries in row n is A051286(n).
T(n,0)=A098479(n).
Sum(k*T(n,k), k=0..n)=A182886(n).

			T(3,1)=2. Indeed, denoting by h (H) the (1,0)-step of weight 1 (2), and u=(1,1), d=(1,-1), the five paths of weight 3 are ud, du, hH, Hh, and hhh; two of them have exactly one H step.
Triangle starts:
1;
1;
1,1;
3,2;
7,3,1;
13,10,3;
27,29,6,1;

M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306.
E. Munarini, N. Zagaglia Salvi, On the rank polynomial of the lattice of order ideals of fences and crowns, Discrete Mathematics 259 (2002), 163-177.

Cf. A051286, A098479, A182886.

G:=1/sqrt(1-2*z-2*t*z^2+z^2+2*t*z^3+t^2*z^4-4*z^3): Gser:=simplify(series(G,z=0,18)): for n from 0 to 14 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 0 to 14 do seq(coeff(P[n],t,k),k=0..floor(n/2)) od; # yields sequence in triangular form

G.f.: G(t,z) =1/sqrt(1-2z-2tz^2+z^2+2t*z^3+t^2*z^4-4z^3).

cp's OEIS Frontend

A182887 Number of (1,0)-steps in all weighted lattice paths in L_n.

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