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A247293 Number of weighted lattice paths B(n) having no uHd strings.

%I A247293 #16 Jul 26 2022 14:45:13
%S A247293 1,1,2,4,8,16,35,77,172,391,899,2085,4877,11490,27236,64916,155483,
%T A247293 374027,903286,2189219,5322965,12980660,31740404,77804885,191160040,
%U A247293 470662449,1161123461,2869754099,7104856781,17618234456,43754467510,108816781175
%N A247293 Number of weighted lattice paths B(n) having no uHd strings.
%C A247293 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 A247293 a(n) = A247292(n,0).
%H A247293 Alois P. Heinz, <a href="/A247293/b247293.txt">Table of n, a(n) for n = 0..1000</a>
%H A247293 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 A247293 G.f. G = G(z) satisfies G = 1 + z*G + z^2*G + z^3*G*(G - z^2).
%F A247293 D-finite with recurrence +(n+3)*a(n) +(-2*n-3)*a(n-1) -n*a(n-2) +(-2*n+3)*a(n-3) +(n-3)*a(n-4) +(2*n-9)*a(n-5) +2*(-n+6)*a(n-6) +(-2*n+15)*a(n-7) +(n-12)*a(n-10)=0. - _R. J. Mathar_, Jul 26 2022
%e A247293 a(6)=35 because among the 37 (=A004148(7)) members of B(6) only huHd and uHdh contain uHd.
%p A247293 eq := G = 1+z*G+z^2*G+z^3*(G-z^2)*G: G := RootOf(eq, G): Gser := series(G, z = 0, 37): seq(coeff(Gser, z, n), n = 0 .. 35);
%p A247293 # second Maple program:
%p A247293 b:= proc(n, y, t) option remember; `if`(y<0 or y>n or t=3, 0,
%p A247293       `if`(n=0, 1, b(n-1, y, 0)+`if`(n>1, b(n-2, y, `if`(t=1,
%p A247293       2, 0))+b(n-2, y+1, 1), 0)+b(n-1, y-1, `if`(t=2, 3, 0))))
%p A247293     end:
%p A247293 a:= n-> b(n, 0$2):
%p A247293 seq(T(n), n=0..40);  # _Alois P. Heinz_, Sep 16 2014
%t A247293 b[n_, y_, t_] := b[n, y, t] = If[y<0 || y>n || t == 3, 0, If[n == 0, 1, b[n-1, y, 0] + If[n>1, b[n-2, y, If[t == 1, 2, 0]] + b[n-2, y+1, 1], 0] + b[n-1, y-1, If[t == 2, 3, 0]]]]; a[n_] := b[n, 0, 0]; Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, May 27 2015, after _Alois P. Heinz_ *)
%Y A247293 Cf. A004148, A247291, A247292, A247295.
%K A247293 nonn
%O A247293 0,3
%A A247293 _Emeric Deutsch_, Sep 16 2014