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A129165 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having k base pyramids.

%I A129165 #7 Jul 23 2017 12:22:46
%S A129165 1,0,1,1,1,1,5,2,2,1,19,9,4,3,1,73,37,15,7,4,1,292,147,63,24,11,5,1,
%T A129165 1203,598,258,100,37,16,6,1,5065,2497,1067,419,152,55,22,7,1,21697,
%U A129165 10633,4507,1762,647,224,79,29,8,1,94274,45980,19379,7528,2765,964,322
%N A129165 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having k base pyramids.
%C A129165 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. A pyramid in a skew Dyck word (path) is a factor of the form u^h d^h, h being the height of the pyramid. A base pyramid is a pyramid starting on the x-axis.
%C A129165 Row sums yield A002212.
%H A129165 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 A129165 T(n,0) = A129166(n).
%F A129165 Sum_{k=0..n} k*T(n,k) = A129167(n).
%F A129165 G.f.: G(t,z) = (1-z)(1 - z + zg)/(1 - z(1-z)g - tz), where g = 1 + zg^2 + z(g-1) = (1 - z - sqrt(1 - 6z + 5z^2))(2z).
%e A129165 T(3,1)=2 because we have (UD)UUDL and (UUUDDD) (the base pyramids are shown between parentheses).
%e A129165 Triangle starts:
%e A129165    1;
%e A129165    0, 1;
%e A129165    1, 1, 1;
%e A129165    5, 2, 2, 1;
%e A129165   19, 9, 4, 3, 1;
%p A129165 g:=(1-z-sqrt(1-6*z+5*z^2))/2/z: G:=(1-z)*(1-z+z*g)/(1-z*(1-z)*g-t*z): Gser:=simplify(series(G,z=0,15)): for n from 0 to 11 do P[n]:=sort(expand(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 A129165 Cf. A002212, A129166, A129167.
%K A129165 nonn,tabl
%O A129165 0,7
%A A129165 _Emeric Deutsch_, Apr 04 2007