A247300 Number of h- and H-steps at level 0 in all lattice paths in B(n).
0, 1, 3, 7, 17, 40, 94, 222, 526, 1252, 2994, 7191, 17343, 41989, 102023, 248712, 608168, 1491349, 3666685, 9037003, 22323243, 55259206, 137058248, 340567477, 847711177, 2113455657, 5277115687, 13195311961, 33038994039, 82829585094, 207905352180
Offset: 0
Keywords
Examples
a(3)=7 because in B(3) = {ud, hH, Hh, hhh} all h- and H-steps are at level 0.
Links
- Alois P. Heinz, 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.
Crossrefs
Cf. A247299.
Programs
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Maple
G := 4*z*(1+z)/(1-z-z^2+sqrt((1+z+z^2)*(1-3*z+z^2)))^2: Gser := series(G, z = 0, 33): seq(coeff(Gser, z, n), n = 0 .. 30); # second Maple program: b:= proc(n, y) option remember; `if`(y<0 or y>n or n<0, 0, `if`(n=0, [1, 0], (p-> p+`if`(y=0, [0, p[1]], 0)) (b(n-1, y) +b(n-2, y)) +b(n-2, y+1) +b(n-1, y-1))) end: a:= n-> b(n, 0)[2]: seq(a(n), n=0..50); # Alois P. Heinz, Sep 17 2014 -
Mathematica
b[n_, y_] := b[n, y] = If[y<0 || y>n || n<0, 0, If[n == 0, {1, 0}, Function[{p}, p + If[y == 0, {0, p[[1]]}, 0]][b[n-1, y] + b[n-2, y]] + b[n-2, y+1] + b[n-1, y-1]]] ; a[n_] := b[n, 0][[2]]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, May 27 2015, after Alois P. Heinz *)
Formula
G.f.: 4*z*(1 + z)/(1 - z - z^2 +sqrt((1 + z + z^2)*(1 - 3*z + z^2)))^2.
a(n) ~ sqrt(525 + 235*sqrt(5)) * ((3 + sqrt(5))/2)^n / (sqrt(2*Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 06 2016
Equivalently, a(n) ~ 5^(3/4) * phi^(2*n + 4) / (sqrt(Pi) * n^(3/2)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 06 2021
D-finite with recurrence -(n+5)*(74*n-105)*a(n) +(90*n^2+287*n+109)*a(n-1) +(190*n^2-89*n+43)*a(n-2) +(206*n^2-289*n+23)*a(n-3) +(42*n^2-301*n+337)*a(n-4) -(58*n-37)*(n-5)*a(n-5)=0. - R. J. Mathar, Jul 24 2022
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