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%I #10 Jul 23 2017 12:17:34
%S 1,0,1,2,0,1,4,5,0,1,14,12,9,0,1,44,53,25,14,0,1,150,196,132,44,20,0,
%T 1,520,777,555,269,70,27,0,1,1850,3064,2486,1260,485,104,35,0,1,6696,
%U 12233,10902,6264,2496,804,147,44,0,1,24602,49096,47955,30108,13600
%N Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n having k ascents of length 1.
%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 a path is defined to be the number of its steps. An ascent in a path is a maximal sequence of consecutive U steps.
%C Row sums yield A002212.
%H E. Deutsch, E. Munarini, 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.
%F T(n,0) = A128750(n).
%F Sum_{k=0..n} k*T(n,k) = A085362(n-1).
%F G.f.: G = G(t,z) satisfies z(1 + z - tz)G^2 - (1 - tz + tz^2 - z^2)G + 1 - z = 0.
%e T(3,1)=5 because we have (U)DUUDD, (U)DUUDL, UUDD(U)D, UUD(U)DD and UUD(U)DL (the ascents of length 1 are shown between parentheses).
%e Triangle starts:
%e 1;
%e 0, 1;
%e 2, 0, 1;
%e 4, 5, 0, 1;
%e 14, 12, 9, 0, 1;
%e 44, 53, 25, 14, 0, 1;
%p eq:=z*(1+z-t*z)*G^2-(1-t*z+t*z^2-z^2)*G+1-z=0: G:=RootOf(eq,G): Gser:=simplify(series(G,z=0,15)): for n from 0 to 11 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 0 to 11 do seq(coeff(P[n],t,j),j=0..n) od; # yields sequence in triangular form
%Y Cf. A002212, A085362, A128750.
%K tabl,nonn
%O 0,4
%A _Emeric Deutsch_, Mar 31 2007
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