A182888 -id:A182888 - cp's OEIS frontend

A182890 Number of (1,0)-steps of weight 1 at level 0 in all weighted lattice paths in L_n.

0, 1, 2, 5, 14, 36, 94, 247, 646, 1691, 4428, 11592, 30348, 79453, 208010, 544577, 1425722, 3732588, 9772042, 25583539, 66978574, 175352183, 459077976, 1201881744, 3146567256, 8237820025, 21566892818, 56462858429, 147821682470, 387002188980, 1013184884470

Offset: 0

Emeric Deutsch, Dec 12 2010

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.

			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.

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 and N. Zagaglia Salvi, On the Rank Polynomial of the Lattice of Order Ideals of Fences and Crowns, Discrete Mathematics 259 (2002), 163-177.
Index entries for linear recurrences with constant coefficients, signature (2,1,2,-1).

Cf. A182888.

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);

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

a(n):=1/2*sum(binomial(2*n-2*m, 2*m+1), m, 0, (2*n-1)/4); /* Vladimir Kruchinin, Jan 24 2022 */

G.f: x/((1+x+x^2)*(1-3*x+x^2)).
a(n) = Sum_{k>=0} k*A182888(n,k).
a(n) = (A000045(2n+2) - ((-1)^n)*A010892(n))/4. - John M. Campbell, Dec 30 2016
a(n) = Sum_{m=0..n} C(2*n-2*m,2*m+1)/2. - Vladimir Kruchinin, Jan 24 2022

A182889 Number of weighted lattice paths in L_n having no (1,0)-steps at level 0. The members of L_n are paths of weight n 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, and 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, 0, 1, 2, 3, 8, 17, 38, 89, 206, 485, 1152, 2751, 6614, 15983, 38798, 94569, 231342, 567771, 1397562, 3449285, 8533886, 21161001, 52579900, 130896887, 326440746, 815437967, 2040049514, 5111051473, 12822135138, 32207384995, 80995950182, 203917464635

Offset: 0

Emeric Deutsch, Dec 11 2010

nonn

a(n)=A182888(n,0).

			a(3)=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, namely ud and du, have no h steps at level 0.

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. A182888.

G:=1/(z+sqrt((1+z+z^2)*(1-3*z+z^2))): Gser:=series(G,z=0,35): seq(coeff(Gser,z,n),n=0..32);

CoefficientList[Series[1/(x+Sqrt[(1+x+x^2)(1-3x+x^2)]),{x,0,40}],x] (* Harvey P. Dale, Jun 16 2013 *)

G.f.: G(z) =1/( z+sqrt((1+z+z^2)(1-3z+z^2)) ).
a(n) ~ sqrt(7*sqrt(5)-15) * ((3 + sqrt(5))/2)^(n+2) / (sqrt(2*Pi) * n^(3/2)). - Vaclav Kotesovec, Mar 06 2016
Equivalently, a(n) ~ 5^(1/4) * phi^(2*n + 2) / (sqrt(Pi) * n^(3/2)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 06 2021
D-finite with recurrence n*a(n) +(-4*n+3)*a(n-1) +(n-3)*a(n-2) +(2*n-3)*a(n-3) +12*(n-3)*a(n-4) +(2*n-9)*a(n-5) +(n-3)*a(n-6) +(-4*n+21)*a(n-7) +(n-6)*a(n-8)=0. - R. J. Mathar, Jul 22 2022

cp's OEIS Frontend

A182890 Number of (1,0)-steps of weight 1 at level 0 in all weighted lattice paths in L_n.

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