A129164 -id:A129164 - cp's OEIS frontend

A129163 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and pyramid weight k.

1, 1, 2, 2, 4, 4, 4, 11, 13, 8, 8, 29, 46, 38, 16, 16, 74, 150, 167, 104, 32, 32, 184, 461, 652, 554, 272, 64, 64, 448, 1354, 2344, 2535, 1724, 688, 128, 128, 1072, 3836, 7922, 10462, 9103, 5112, 1696, 256, 256, 2528, 10552, 25506, 40007, 42547, 30773, 14592

Offset: 1

Emeric Deutsch, Apr 03 2007

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 pyramid in a skew Dyck word w is maximal if, as a factor in w, it is not immediately preceded by a U and immediately followed by a D. The pyramid weight of a skew Dyck path (word) is the sum of the heights of its maximal pyramids.

Row sums yield A002212. T(n,1)=2^(n-2) (n>=2). T(n,n)=2^(n-1). Sum(k*T(n,k),k=1..n)=A129164(n). Pyramid weight in Dyck paths is considered in the Denise and Simion reference (see also A091866).

			T(3,2)=4 because we have (UD)U(UD)L, U(UD)(UD)D, U(UD)(UD)L and U(UUDD)L (the maximal pyramids are shown between parentheses).
Triangle starts:
1;
1,2;
2,4,4;
4,11,13,8;
8,29,46,38,16;

A. Denise and R. Simion, Two combinatorial statistics on Dyck paths, Discrete Math., 137, 1995, 155-176.
E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203

Cf. A002212, A129164.

eq:=z*(1-t*z)*G^2-(1-2*t*z+t*z^2)*G+(1-z)*(1-t*z)=0: G:=RootOf(eq,G): Gser:=simplify(series(G-1,z=0,15)): for n from 1 to 11 do P[n]:=sort(expand(coeff(Gser,z,n))) od: for n from 1 to 11 do seq(coeff(P[n],t,j),j=1..n) od; # yields sequence in triangular form

G.f.=G-1, where G=G(t,z) is given by z(1-tz)G^2-(1-2tz+tz^2)G+(1-z)(1-tz)=0.

A246657 a(n) = round(3F2([1, 3/2, 1 - n], [2, 2], -4)).

Original entry on oeis.org

0, 1, 3, 7, 24, 87, 332, 1320, 5407, 22672, 96844, 419910, 1843386, 8176962, 36594388, 165026353, 749170529, 3420949803, 15702232962, 72407225094, 335276107549, 1558289108596, 7267201176311, 33996105203757, 159484982985619, 750134031377432

Offset: 0

Peter Luschny, Sep 16 2014

nonn

Without rounding, the sequence were 1, 5/2, 22/3, 97/4, 436/5, 997/3, 9241/7, 43257/8,... for n>=1. - R. J. Mathar, Jul 27 2022

Cf. A129164.

A246657 = lambda n: hypergeometric([1, 3/2, 1-n], [2, 2], -4)
[round(A246657(n).n(100)) for n in (0..25)]

a(n) = A129164(n)/n for n>=1.

cp's OEIS Frontend

A129163 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and pyramid weight k.

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