A247295 - OEIS

%I #13 Jul 26 2022 14:48:04

%S 1,1,2,4,7,14,30,64,141,316,713,1626,3740,8659,20176,47274,111302,

%T 263201,624860,1488736,3558412,8530533,20505468,49413242,119347708,

%U 288873639,700582008,1702190653,4142880297,10099352082,24656876772,60283224645,147581756005

%N Number of weighted lattice paths B(n) having no uhd and no uHd strings.

%C B(n) is the set of lattice paths of weight n that start in (0,0), end on the horizontal axis and never go below this axis, whose steps are of the following four kinds: h = (1,0) of weight 1, H = (1,0) of weight 2, u = (1,1) of weight 2, and d = (1,-1) of weight 1. The weight of a path is the sum of the weights of its steps.

%C a(n) = A247294(n,0).

%H Alois P. Heinz, <a href="/A247295/b247295.txt">Table of n, a(n) for n = 0..1000</a>

%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.

%F G.f. G = G(z) satisfies G = 1 + z*G + z^2*G + z^3*G*(G - z- z^2 ).

%F D-finite with recurrence (n+3)*a(n) +(-2*n-3)*a(n-1) -n*a(n-2) +(-2*n+3)*a(n-3) +3*(n-3)*a(n-4) +4*(-n+6)*a(n-6) +(-2*n+15)*a(n-7) +(n-9)*a(n-8) +(2*n-21)*a(n-9) +(n-12)*a(n-10)=0. - _R. J. Mathar_, Jul 26 2022

%e a(6)=30 because among the 37 (=A004148(7)) members of B(6) only uhdhh, huhdh, hhuhd, Huhd, uhdH, uHdh, and huHd contain uhd or uHd (or both).

%p eq := G = 1+z*G+z^2*G+z^3*(G-z-z^2)*G: G := RootOf(eq, G): Gser := series(G, z = 0, 37): seq(coeff(Gser, z, n), n = 0 .. 35);

%p # second Maple program:

%p b:= proc(n, y, t) option remember; `if`(y<0 or y>n or t=3, 0,

%p       `if`(n=0, 1, b(n-1, y-1, `if`(t=2, 3, 0))+b(n-1, y,

%p       `if`(t=1, 2, 0))+`if`(n>1, b(n-2, y, `if`(t=1, 2, 0))+

%p        b(n-2, y+1, 1), 0)))

%p     end:

%p a:= n-> b(n, 0$2):

%p seq(a(n), n=0..40);  # _Alois P. Heinz_, Sep 16 2014

%t b[n_, y_, t_] := b[n, y, t] = If[y<0 || y>n || t == 3, 0, If[n == 0, 1, b[n-1, y-1, If[t == 2, 3, 0]] + b[n-1, y, If[t == 1, 2, 0]] + If[n>1, b[n-2, y, If[t == 1, 2, 0]] + b[n-2, y+1, 1], 0]]]; a[n_] := b[n, 0, 0]; Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, May 27 2015, after _Alois P. Heinz_ *)

%Y Cf. A004148, A247291, A247293, A247294.

%K nonn

%O 0,3

%A _Emeric Deutsch_, Sep 16 2014