A247294 Triangle read by rows: T(n,k) is the number of weighted lattice paths B(n) having a total of k uhd and uHd strings.

1, 1, 2, 4, 7, 1, 14, 3, 30, 7, 64, 18, 141, 43, 1, 316, 102, 5, 713, 249, 16, 1626, 608, 49, 3740, 1489, 143, 1, 8659, 3669, 400, 7, 20176, 9058, 1109, 29, 47274, 22407, 3046, 105, 111302, 55560, 8282, 357, 1, 263201, 138004, 22385, 1149, 9

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) = A247295(n).

Sum(k*T(n,k), k=0..n) = A247296(n).

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 - z^2 + t*z + t*z^2).

Example

T(6,1)=7 because we have uhdhh, huhdh, hhuhd, Huhd, uhdH, uHdh, and huHd.
Triangle starts:
1;
1;
2;
4;
7,1;
14,3;
30,7;

Maple

eq := G = 1+z*G+z^2*G+z^3*(G-z-z^2+t*z+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((1/4)*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-1, 0)*`if`(t=2, x, 1)+b(n-1, y, `if`(t=1, 2, 0))
      +`if`(n>1, b(n-2, y, `if`(t=1, 2, 0))+b(n-2, y+1, 1), 0))))
    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-1, 0]*If[t == 2, x, 1] + b[n-1, y, If[t == 1, 2, 0]] + If[n>1, b[n-2, y, If[t == 1, 2, 0]] + b[n-2, y+1, 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, 20}] // Flatten (* Jean-François Alcover, May 27 2015, after Alois P. Heinz *)

Crossrefs

Cf. A004148, A247290, A247292, A247295, A247296.

Sequence in context: A110317 A098073 A118390 * A202848 A202841 A247290

Adjacent sequences: A247291 A247292 A247293 * A247295 A247296 A247297

Keyword

nonn,tabf

Author

Emeric Deutsch, Sep 16 2014

Status

approved