

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.


3



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
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OFFSET

0,3


LINKS



FORMULA

G.f.: G(t,z) = 1/(1zz^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).


EXAMPLE

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;
...


MAPLE

eq := c = 1+z*c+z^2*c+z^3*c^2: c := RootOf(eq, c): G := 1/(1zz^2t*z^3*cz^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


PROG

(PARI)
T(n)={[Vecrev(p)  p<Vec(1/(1xx^2  (1+y)*(1xx^2  sqrt(1+x^42*x^3x^22*x+O(x*x^n)))/2))]}
{ my(A=T(12)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Nov 05 2019


CROSSREFS



KEYWORD

nonn,walk,tabf


AUTHOR



EXTENSIONS



STATUS

approved



