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A247290 Triangle read by rows: T(n,k) is the number of weighted lattice paths B(n) having k uhd strings. 4
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified July 12 22:46 EDT 2020. Contains 335669 sequences. (Running on oeis4.)