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 A129163 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and pyramid weight k. 1
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS 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). LINKS 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 FORMULA 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. EXAMPLE 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; MAPLE 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 CROSSREFS Cf. A002212, A129164. Sequence in context: A220461 A220287 A320195 * A083549 A083548 A082849 Adjacent sequences:  A129160 A129161 A129162 * A129164 A129165 A129166 KEYWORD nonn,tabl AUTHOR Emeric Deutsch, Apr 03 2007 STATUS approved

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Last modified January 20 10:18 EST 2022. Contains 350471 sequences. (Running on oeis4.)