A247292 Triangle read by rows: T(n,k) is the number of weighted lattice paths B(n) having k uHd strings.

1, 1, 2, 4, 8, 16, 1, 35, 2, 77, 5, 172, 13, 391, 32, 899, 78, 1, 2085, 195, 3, 4877, 487, 9, 11490, 1217, 28, 27236, 3055, 81, 64916, 7687, 228, 1, 155483, 19374, 641, 4, 374027, 48925, 1782, 14, 903286, 123760, 4908, 50, 2189219, 313512, 13451, 165, 5322965, 795263, 36690, 522, 1

Offset: 0

Author

Emeric Deutsch, Sep 16 2014

Keywords

• nonn
• tabf

Comments

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.

Row n contains 1 + floor(n/5) entries.

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

T(n,0) = A247293(n).

Sum(k*T(n,k), k=0..n) = A110320(n-4) (n>=4).

Examples

			T(6,1)=2 because we have uHdh and huHd.
Triangle starts:
1;
1;
2;
4;
8;
16,1;
35,2;
		

Links

• Alois P. Heinz, Rows n = 0..320, flattened
• 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. A004148, A110320, A247290, A247293, A247294.

Programs

Maple

eq := G = 1+z*G+z^2*G+z^3*(G-z^2+t*z^2)*G: G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 25)): for n from 0 to 22 do P[n] := sort(coeff(Gser, z, n)) end do; for n from 0 to 22 do seq(coeff(P[n], t, k), k = 0 .. floor((1/5)*n)) 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, 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, 0)*`if`(t=2, x, 1))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0$2)):
seq(T(n), n=0..20);  # Alois P. Heinz, Sep 16 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, 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, 0]*If[t == 2, x, 1]]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 0, 0]]; Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, May 27 2015, after Alois P. Heinz *)

Formula

G.f. G = G(t,z) satisfies G = 1 + z*G + z^2*G + z^3*G*(G - z^2 + t*z^2).