cp's OEIS Frontend

A274372 Number T(n,k) of modified skew Dyck paths of semilength n such that the area between the x-axis and the path is k; triangle T(n,k), n>=0, n<=k<=n^2, read by rows.

%I A274372 #22 Jan 25 2017 03:05:55
%S A274372 1,1,1,0,1,1,0,2,1,1,0,1,1,0,3,2,3,1,3,2,2,1,1,0,1,1,0,4,3,7,4,7,5,8,
%T A274372 6,6,3,5,4,3,2,2,1,1,0,1,1,0,5,4,12,10,17,12,20,18,22,14,19,16,18,14,
%U A274372 14,12,12,7,8,7,5,4,3,2,2,1,1,0,1
%N A274372 Number T(n,k) of modified skew Dyck paths of semilength n such that the area between the x-axis and the path is k; triangle T(n,k), n>=0, n<=k<=n^2, read by rows.
%C A274372 A modified 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 A=(-1,1) (anti-down) so that A and D steps do not overlap.
%H A274372 Alois P. Heinz, <a href="/A274372/b274372.txt">Rows n = 0..40, flattened</a>
%F A274372 Sum_{k=n..n^2} k * T(n,k) = A274373(n).
%F A274372 T(n,n) = T(n,n^2) = 1.
%F A274372 T(n,n+1) = T(n,n^2-1) = 0.
%F A274372 T(n,n*(n+1)/2) = T(n,A000217(n)) = A274054(n).
%e A274372 T(3,3) = 1:   /\/\/\
%e A274372 .
%e A274372                  /\         /\
%e A274372 T(3,5) = 2:   /\/  \   ,   /  \/\
%e A274372 .
%e A274372               /\
%e A274372               \ \
%e A274372 T(3,6) = 1:   /  \
%e A274372 .
%e A274372                /\/\
%e A274372 T(3,7) = 1:   /    \
%e A274372 .
%e A274372                 /\
%e A274372                /  \
%e A274372 T(3,9) = 1:   /    \
%e A274372 .
%e A274372 Triangle T(n,k) begins:
%e A274372 n\k: 0 1 2 3 4 5 6 7 8 9 . . . . . .16 . . . . . . . .25
%e A274372 ---+----------------------------------------------------
%e A274372 00 : 1
%e A274372 01 :   1
%e A274372 02 :     1 0 1
%e A274372 03 :       1 0 2 1 1 0 1
%e A274372 04 :         1 0 3 2 3 1 3 2 2 1 1 0 1
%e A274372 05 :           1 0 4 3 7 4 7 5 8 6 6 3 5 4 3 2 2 1 1 0 1
%p A274372 b:= proc(x, y, t, n) option remember; expand(`if`(y>n, 0,
%p A274372       `if`(n=y, `if`(t=2, 0, 1), b(x+1, y+1, 0, n-1)*z^
%p A274372       (2*y+1)+`if`(t<>1 and x>0, b(x-1, y+1, 2, n-1), 0)+
%p A274372       `if`(t<>2 and y>0, b(x+1, y-1, 1, n-1), 0))))
%p A274372     end:
%p A274372 T:= n-> (p-> seq(coeff(p, z, i), i=n..n^2))(b(0$3, 2*n)):
%p A274372 seq(T(n), n=0..8);
%t A274372 b[x_, y_, t_, n_] := b[x, y, t, n] = Expand[If[y>n, 0, If[n == y, If[t == 2, 0, 1], b[x+1, y+1, 0, n-1]*z^(2*y+1) + If[t != 1 && x>0, b[x-1, y+1, 2, n-1], 0] + If[t != 2 && y>0, b[x+1, y-1, 1, n-1], 0]]]]; T[n_] := Function[p, Table[Coefficient[p, z, i], {i, n, n^2}]][b[0, 0, 0, 2*n]]; Table[T[n], {n, 0, 8}] // Flatten (* _Jean-François Alcover_, Jan 25 2017, translated from Maple *)
%Y A274372 Row sums give: A230823.
%Y A274372 Column sums give: A274376.
%Y A274372 Cf. A000217, A002061 (number of terms in row n), A129172, A274054, A274373.
%K A274372 nonn,tabf
%O A274372 0,8
%A A274372 _Alois P. Heinz_, Jun 19 2016