A246182 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.

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

Offset

0,5

Comments

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

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

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

Links

Alois P. Heinz, Rows n = 0..141, flattened

M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306.

Formula

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.

Example

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.
Triangle starts:
1;
1;
1,1;
3,0,1;
5,2,0,1;
9,5,2,0,1;

Maple

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
# second Maple program:
b:= proc(n, y, t) option remember; `if`(y<0 or 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))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0$2)):
seq(T(n), n=0..12); # Alois P. Heinz, Aug 24 2014

Mathematica

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 *)

Crossrefs

Cf. A004148, A110320, A246177.

Sequence in context: A115714 A346242 A020768 * A104544 A123880 A186363

Adjacent sequences: A246179 A246180 A246181 * A246183 A246184 A246185

Keyword

nonn,tabf

Author

Emeric Deutsch, Aug 23 2014

Status

approved