A247290 Triangle read by rows: T(n,k) is the number of weighted lattice paths B(n) having k uhd strings.

1, 1, 2, 4, 7, 1, 15, 2, 32, 5, 69, 13, 154, 30, 1, 346, 74, 3, 786, 183, 9, 1806, 449, 28, 4180, 1114, 78, 1, 9745, 2767, 219, 4, 22865, 6882, 611, 14, 53938, 17170, 1674, 50, 127865, 42906, 4569, 161, 1, 304447, 107392, 12398, 506, 5, 727733, 269237, 33450, 1564, 20

Offset

0,3

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/4) entries.

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

T(n,0) = A247291(n).

Sum(k*T(n,k), k=0..n) = A110320(n-3) (n>=3)

Links

Alois P. Heinz, Rows n = 0..300, 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 G = 1 + z*G + z^2*G + z^3*G*(G - z + t*z).

Example

T(5,1)=2 because we have huhd and uhdh.
Triangle starts:
1;
1;
2;
4;
7,1;
15,2;

Maple

eq := G = 1+z*G+z^2*G+z^3*(G-z+t*z)*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((1/4)*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, `if`(t=1, 2, 0))+`if`(n>1, b(n-2, y, 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, If[t == 1, 2, 0]] + If[n>1, b[n-2, y, 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 *)

Crossrefs

Cf. A004148, A110320, A247291, A247292, A247294.

Sequence in context: A247294 A202848 A202841 * A246183 A134974 A244262

Adjacent sequences: A247287 A247288 A247289 * A247291 A247292 A247293

Keyword

nonn,tabf

Author

Emeric Deutsch, Sep 16 2014

Status

approved