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%I #15 Mar 27 2021 08:07:46
%S 1,1,2,5,1,14,6,42,28,3,132,120,28,1,429,495,180,20,1430,2002,990,195,
%T 10,4862,8008,5005,1430,165,4,16796,31824,24024,9009,1650,117,1,58786,
%U 125970,111384,51688,13013,1617,70,208012,497420,503880,278460,89180,16016,1386,35
%N Number T(n,k) of modified skew Dyck paths of semilength n with exactly k anti-down steps; triangle T(n,k), n>=0, 0<=k<=n-floor((1+sqrt(max(0,8n-7)))/2), read by rows.
%C 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 Alois P. Heinz, <a href="/A274404/b274404.txt">Rows n = 0..160, flattened</a>
%F Sum_{k>0} k * T(n,k) = A274405(n).
%e /\
%e \ \
%e T(3,1) = 1: / \
%e .
%e Triangle T(n,k) begins:
%e : 1;
%e : 1;
%e : 2;
%e : 5, 1;
%e : 14, 6;
%e : 42, 28, 3;
%e : 132, 120, 28, 1;
%e : 429, 495, 180, 20;
%e : 1430, 2002, 990, 195, 10;
%e : 4862, 8008, 5005, 1430, 165, 4;
%e : 16796, 31824, 24024, 9009, 1650, 117, 1;
%p b:= proc(x, y, t, n) option remember; expand(`if`(y>n, 0,
%p `if`(n=y, `if`(t=2, 0, 1), b(x+1, y+1, 0, n-1)+
%p `if`(t<>1 and x>0, b(x-1, y+1, 2, n-1)*z, 0)+
%p `if`(t<>2 and y>0, b(x+1, y-1, 1, n-1), 0))))
%p end:
%p T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(0$3, 2*n)):
%p seq(T(n), n=0..14);
%t b[x_, y_, t_, n_] := b[x, y, t, n] = Expand[If[y > n, 0,
%t If[n == y, If[t == 2, 0, 1], b[x + 1, y + 1, 0, n - 1] +
%t If[t != 1 && x > 0, b[x - 1, y + 1, 2, n - 1] z, 0] +
%t If[t != 2 && y > 0, b[x + 1, y - 1, 1, n - 1], 0]]]];
%t T[n_] := CoefficientList[b[0, 0, 0, 2n], z];
%t T /@ Range[0, 14] // Flatten (* _Jean-François Alcover_, Mar 27 2021, after _Alois P. Heinz_ *)
%Y Columns k=0-3 give: A000108, A002694(n-1), A074922(n-2), A232224(n-3).
%Y Row sums give A230823.
%Y Last elements of rows give A092392(n-1) for n>0.
%Y Cf. A083920, A274405.
%K nonn,tabf
%O 0,3
%A _Alois P. Heinz_, Jun 20 2016
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