

A094449


Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n and having sum of pyramid heights equal to k.


0



1, 0, 1, 0, 0, 2, 1, 0, 0, 4, 4, 2, 0, 0, 8, 13, 8, 5, 0, 0, 16, 42, 26, 20, 12, 0, 0, 32, 139, 85, 65, 48, 28, 0, 0, 64, 470, 286, 214, 156, 112, 64, 0, 0, 128, 1616, 982, 727, 517, 364, 256, 144, 0, 0, 256, 5632, 3420, 2518, 1772, 1214, 832, 576, 320, 0, 0, 512, 19852
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OFFSET

0,6


COMMENTS

A pyramid in a Dyck path is a factor of the form U^j D^j (j>0), starting at the xaxis. Here U=(1,1) and D(1,1). This definition differs from the one in A091866. Column k=0 is A082989. Row sums are the Catalan numbers (Q000108).


LINKS



FORMULA

G.f.=G=G(t, z)= (1tz)(1z)/[12tz+tz^2z(1z)(1t*z)C), where C=[1sqrt(14z)]/(2z) is the Catalan function.


EXAMPLE

T(3,3)=4 because there are four Dyck paths of semilength 3 having 3 as sum of pyramid heights: (UD)(UUDD),(UUDD)(UD),(UD)(UD)(UD) and (UUUDDD) (the pyramids are shown between parentheses).
Triangle begins:
[1];[0, 1];[0, 0, 2];[1, 0, 0, 4];[4, 2, 0, 0, 8];[13, 8, 5, 0, 0, 16];[42, 26, 20, 12, 0, 0, 32];


MAPLE

C:=(1sqrt(14*z))/2/z: G:=(1t*z)*(1z)/(12*t*z+t*z^2z*C*(1z)*(1t*z)): Gserz:=simplify(series(G, z=0, 16)): P[0]:=1: for n from 1 to 14 do P[n]:=sort(coeff(Gserz, z^n)) od: seq([subs(t=0, P[n]), seq(coeff(P[n], t^k), k=1..n)], n=0..14);


CROSSREFS



KEYWORD



AUTHOR



STATUS

approved



