A246182 - OEIS

%I #17 Jun 29 2015 08:31:08

%S 1,1,1,1,3,0,1,5,2,0,1,9,5,2,0,1,19,9,6,2,0,1,39,21,12,7,2,0,1,79,53,

%T 27,15,8,2,0,1,167,118,74,34,18,9,2,0,1,357,269,180,96,42,21,10,2,0,1,

%U 763,639,419,254,119,51,24,11,2,0,1,1651,1486,1045,605,340,143,61,27,12,2,0,1

%N Triangle read by rows: T(n,k) is the number of weighted lattice paths B(n) having k hh's. 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: a (1,0)-step h of weight 1; a (1,0)-step H of weight 2; a (1,1)-step u of weight 2; a (1,-1)-step d of weight 1. The weight of a path is the sum of the weights of its steps.

%C Number of entries in row n is n (n>=1).

%C Sum of entries in row n is A004148(n+1) (the 2ndary structure numbers).

%C Sum(k*T(n,k), k>=0) = A110320(n-1) (n>=1).

%H Alois P. Heinz, <a href="/A246182/b246182.txt">Rows n = 0..141, flattened</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(t,z) satisfies z^3*(1+z-t*z)*G^2 - (1-t*z-z^2+t*z^3-z^3)*G+1+z-t*z=0.

%e Row 3 is 3,0,1. Indeed, the four paths of weight 3 are: ud, hH, Hh, and hhh, having 0, 0, 0, and 2 hh's, respectively.

%e Triangle starts:

%e 1;

%e 1;

%e 1,1;

%e 3,0,1;

%e 5,2,0,1;

%e 9,5,2,0,1;

%p eq := z^3*(1+z-t*z)*G^2-(-z^3+1-z^2-t*z+t*z^3)*G+1+z-t*z = 0: g := RootOf(eq, G): gser := simplify(series(g, z = 0, 18)): for j from 0 to 15 do P[j] := coeff(gser, z, j) end do: 1; for j to 13 do seq(coeff(P[j], t, q), q = 0 .. j-1) end do; # yields sequence in triangular form

%p # second Maple program:

%p b:= proc(n, y, t) option remember; `if`(y<0 or y>n, 0, `if`(n=0, 1,

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

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

%p     end:

%p T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0$2)):

%p seq(T(n), n=0..12); # _Alois P. Heinz_, Aug 24 2014

%t b[n_, y_, t_] := b[n, y, t] = If[y<0 || y>n, 0, If[n==0, 1, Expand[b[n-1, y, 1] * If[t==1, x, 1] + If[n>1, b[n-2, y, 0] + b[n-2, y+1, 0], 0] + b[n-1, y-1, 0]]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 0, 0]]; Table[T[n], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Jun 29 2015, after _Alois P. Heinz_ *)

%Y Cf. A004148, A110320, A246177.

%K nonn,tabf

%O 0,5

%A _Emeric Deutsch_, Aug 23 2014