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A246183 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, 2, 4, 7, 1, 15, 2, 33, 3, 1, 71, 9, 2, 158, 23, 3, 1, 357, 54, 10, 2, 812, 136, 26, 3, 1, 1869, 338, 63, 11, 2, 4338, 835, 167, 29, 3, 1, 10134, 2087, 428, 72, 12, 2, 23829, 5216, 1092, 199, 32, 3, 1, 56341, 13046, 2826, 523, 81, 13, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Number of entries in row n is floor(n/2) (n>=2).

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

Sum(k*T(n,k), k>=0) = A110320(n-3) (n>=4).

LINKS

Alois P. Heinz, Rows n = 0..200, 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^2-t*z^2)*G^2 - (1-z-t*z^2+t*z^3-z^3)*G +1+z^2-t*z^2=0.

EXAMPLE

Row 3 is 4. Indeed, the four paths of weight 3 are: ud, hH, Hh, and hhh; none of them contain HH.

Triangle starts:

1;

1;

2;

4;

7,1;

15,2;

33,3,1;

MAPLE

eq := z^3*(1+z^2-t*z^2)*G^2-(1-z-t*z^2+t*z^3-z^3)*G+1+z^2-t*z^2 = 0: g := RootOf(eq, G): gser := simplify(series(g, z = 0, 22)): for j from 0 to 20 do P[j] := sort(coeff(gser, z, j)) end do: 1; for j to 20 do seq(coeff(P[j], t, q), q = 0 .. (1/2)*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, 0)+ `if`(n>1, b(n-2, y, 1)*`if`(t=1, x, 1)+

      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..20); # 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, 0] + If[n>1, b[n-2, y, 1]*If[t==1, x, 1] + 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, 20}] // Flatten (* Jean-Fran├žois Alcover, Feb 08 2017, after Alois P. Heinz *)

CROSSREFS

Cf. A004148, A110320.

Sequence in context: A202848 A202841 A247290 * A134974 A244262 A166531

Adjacent sequences:  A246180 A246181 A246182 * A246184 A246185 A246186

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, Aug 23 2014

STATUS

approved

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Last modified July 12 05:35 EDT 2020. Contains 335658 sequences. (Running on oeis4.)