A182896 Triangle read by rows: T(n,k) is the number of weighted lattice paths in L_n having k (1,-1)-returns to the horizontal axis. The members of L_n are paths of weight n that start at (0,0), end on the horizontal axis and whose steps are of the following four kinds: an (1,0)-step with weight 1, an (1,0)-step with weight 2, a (1,1)-step with weight 2, and a (1,-1)-step with weight 1. The weight of a path is the sum of the weights of its steps.

1, 1, 2, 4, 1, 8, 3, 17, 9, 37, 25, 1, 82, 66, 5, 185, 171, 20, 423, 437, 70, 1, 978, 1107, 225, 7, 2283, 2790, 686, 35, 5373, 7009, 2015, 147, 1, 12735, 17574, 5760, 553, 9, 30372, 44019, 16135, 1932, 54, 72832, 110210, 44500, 6398, 264, 1, 175502, 275925, 121247, 20350, 1134, 11

Offset: 0

Author

Emeric Deutsch, Dec 12 2010

Keywords

• nonn
• walk
• tabf

Examples

			T(3,1)=1. Indeed, denoting by h (H) the (1,0)-step of weight 1 (2), and u=(1,1), d=(1,-1), the five paths of weight 3 are ud, du, hH, Hh, and hhh; exactly one of them, namely ud, has one (1,-1)-return to the horizontal axis.
Triangle starts:
   1;
   1;
   2;
   4,  1;
   8,  3;
  17,  9;
  37, 25, 1;
  82, 66, 5;
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1750
• M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306.
• E. Munarini, N. Zagaglia Salvi, On the Rank Polynomial of the Lattice of Order Ideals of Fences and Crowns, Discrete Mathematics 259 (2002), 163-177.

Crossrefs

Cf. A051286, A004148, A182897.

Programs

Maple

eq := c = 1+z*c+z^2*c+z^3*c^2: c := RootOf(eq, c): G := 1/(1-z-z^2-t*z^3*c-z^3*c): Gser := simplify(series(G, z = 0, 19)): for n from 0 to 16 do P[n] := sort(coeff(Gser, z, n)) end do; for n from 0 to 16 do seq(coeff(P[n], t, k), k = 0 .. floor((1/3)*n)) end do: #yields sequence in triangular form

PARI

T(n)={[Vecrev(p) | p<-Vec(1/(1-x-x^2 - (1+y)*(1-x-x^2 - sqrt(1+x^4-2*x^3-x^2-2*x+O(x*x^n)))/2))]}
{ my(A=T(12)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Nov 05 2019

Formula

G.f.: G(t,z) = 1/(1-z-z^2-(1+t)z^3*c), where c satisfies c = 1 + zc + z^2*c + z^3*c^2.

Sum of entries in row n is A051286(n).

T(n,0) = A004148(n+1) (the secondary structure numbers).

Sum_{k=0..n} k*T(n,k) = A182897(n).

Extensions

Data corrected by Andrew Howroyd, Nov 05 2019