A129172 - OEIS

%I #22 Jan 10 2022 10:43:09

%S 1,0,1,0,0,1,1,1,0,0,0,1,1,3,2,1,1,1,0,0,0,0,1,1,4,5,5,3,5,4,3,2,1,1,

%T 1,0,0,0,0,0,1,1,5,7,12,10,11,12,14,12,10,8,10,7,5,4,3,2,1,1,1,0,0,0,

%U 0,0,0,1,1,6,9,19,23,28,26,36,38,38,32,36,36,34,29,27,25,21,15,16,13,10,7,5

%N Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n such that the area between the x-axis and the path is k (n >= 0, 0 <= k <= n^2).

%C A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1)(up), D=(1,-1)(down) and L=(-1,-1)(left) so that up and left steps do not overlap. The length of the path is defined to be the number of its steps.

%C Row n has n^2 + 1 terms, the first n of which are equal to 0.

%C Row sums yield A002212.

%C Sum of terms in column k is the Fibonacci number F(k) (k >= 1; F(1)=1, F(2)=1; A000045).

%H Alois P. Heinz, <a href="/A129172/b129172.txt">Rows n = 0..40, flattened</a>

%H E. Deutsch, E. Munarini and S. Rinaldi, <a href="http://dx.doi.org/10.1016/j.jspi.2010.01.015">Skew Dyck paths</a>, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203.

%H Emeric Deutsch, Emanuele Munarini and Simone Rinaldi, <a href="https://doi.org/10.1016/j.jspi.2009.12.013">Skew Dyck paths, area, and superdiagonal bargraphs</a>, Journal of Statistical Planning and Inference, Vol. 140, Issue 6, June 2010, pp. 1550-1562.

%F Sum_{k=0..n^2} k*T(n,k) = A129173(n).

%F G.f.: G(t,z) = H(t,1,z), where H(t,x,z) = 1+txzH(t,t^2*x,z)H(t,x,z) + z[H(t,t^2*x,z)-1] (H(t,x,z) is the trivariate g.f. for skew Dyck paths according to area, semiabscissa of the last point on the x-axis and semilength, marked by t,x and z, respectively).

%e T(4,7)=5 because we have UDUUUDLD, UDUUDUDL, UUDDUUDL, UUUDLDUD and UUUUDLLL.

%e Triangle starts:

%e   1;

%e   0, 1;

%e   0, 0, 1, 1, 1;

%e   0, 0, 0, 1, 1, 3, 2, 1, 1, 1;

%e   0, 0, 0, 0, 1, 1, 4, 5, 5, 3, 5, 4, 3, 2, 1, 1, 1;

%p G:=(1-z+z*g[1])/(1-t*x*z*g[1]): for i from 1 to 9 do g[i]:=(1-z+z*g[i+1])/(1-t^(2*i+1)*x*z*g[i+1]) od: g[10]:=0: x:=1: Gser:=simplify(series(G,z=0,9)): for n from 0 to 7 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 0 to 7 do seq(coeff(P[n],t,j),j=0..n^2) od; # yields sequence in triangular form

%p # second Maple program:

%p b:= proc(n, y, t) option remember; expand(`if`(y>n, 0,

%p       `if`(n=0, 1, `if`(t<0, 0, b(n-1, y+1, 1)*z^(y+1/2))+

%p       `if`(y<1, 0, b(n-1, y-1, 0)*z^(y-1/2))+

%p       `if`(t>0 or y<1, 0, b(n-1, y-1, -1)*z^(1/2-y)))))

%p     end:

%p T:= n-> (p-> seq(coeff(p, z, i), i=0..n^2))(b(2*n, 0$2)):

%p seq(T(n), n=0..8);  # _Alois P. Heinz_, Jun 19 2016

%t b[n_, y_, t_] := b[n, y, t] = Expand[If[y > n, 0, If[n == 0, 1, If[t < 0, 0, b[n - 1, y + 1, 1]*z^(y + 1/2)] + If[y < 1, 0, b[n - 1, y - 1, 0]*z^(y - 1/2)] + If[t > 0 || y < 1, 0, b[n - 1, y - 1, -1]*z^(1/2 - y)]]]]; T[n_] := Function[p, Table[Coefficient[p, z, i], {i, 0, n^2}]][b[2*n, 0, 0]]; Table[T[n], {n, 0, 8}] // Flatten (* _Jean-François Alcover_, Dec 20 2016, after _Alois P. Heinz_ *)

%Y Cf. A000045, A002212, A129173, A274372.

%K nonn,tabf

%O 0,14

%A _Emeric Deutsch_, Apr 09 2007