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A182906
Triangle read by rows: T(n,k) is the number of weighted lattice paths in F[n] having endpoint height k (k<=floor(n/2)). The members of F[n] are paths of weight n that start at (0,0), do not go below 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.
0
1, 1, 2, 1, 4, 2, 8, 5, 1, 17, 12, 3, 37, 28, 9, 1, 82, 66, 25, 4, 185, 156, 66, 14, 1, 423, 370, 171, 44, 5, 978, 882, 437, 129, 20, 1, 2283, 2112, 1107, 364, 70, 6, 5373, 5079, 2790, 1000, 225, 27, 1, 12735, 12264
OFFSET
0,3
COMMENTS
The paths need not end on the horizontal axis.
Number of entries in row n is 1+floor(n/2).
Sum of entries in row n is A182905(n).
T(n,0) = A004148(n+1).
T(n,1) = A166297(n+1).
LINKS
FORMULA
G.f.: G(t,z) = g/(1-tz^2*g), where g=g(z) is defined by g = 1 + z*g + z^2*g + z^3*g^2.
Rec. rel.: T(n,k) = T(n-1,k) + T(n-1,k+1) + T(n-2,k) + T(n-2,k-1); the 2nd Maple program makes use of this.
T(n,k) = (k+1)*Sum_{i=0..n-k+1} C(i+1,n+1-i)*C(i+1,-i+n-k)/(i+1). - Vladimir Kruchinin, Jan 25 2019
EXAMPLE
T(4,1)=5. Indeed, denoting by h (H) the (1,0)-step of weight 1 (2), and U=(1,1), D=(1,-1), we have hhU, hUh, Uhh, HU, and UH.
Triangle starts:
1;
1;
2, 1;
4, 2;
8, 5, 1;
17, 12, 3;
37, 28, 9, 1;
MAPLE
g := ((1-z-z^2-sqrt((1+z+z^2)*(1-3*z+z^2)))*1/2)/z^3: G := g/(1-t*z^2*g); Gser := simplify(series(G, z = 0, 22)): for n from 0 to 14 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 14 do seq(coeff(P[n], t, k), k = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form
T := proc (n, k) if k < 0 then 0 elif n < 0 then 0 elif (1/2)*n < k then 0 elif n = 0 and k = 0 then 1 else T(n-1, k)+T(n-1, 1+k)+T(n-2, k)+T(n-2, k-1) end if end proc: for n from 0 to 14 do seq(T(n, k), k = 0 .. floor((1/2)*n)) end do;
PROG
(Maxima)
T(n, k):=(k+1)*sum((binomial(i+1, n+1-i)*binomial(i+1, -i+n-k))/(i+1), i, 0, n-k+1);
/* Vladimir Kruchinin, Jan 25 2019 */
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Dec 17 2010
STATUS
approved