This site is supported by donations to The OEIS Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A091869 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having k peaks at even height. 6
 1, 1, 1, 2, 2, 1, 4, 6, 3, 1, 9, 16, 12, 4, 1, 21, 45, 40, 20, 5, 1, 51, 126, 135, 80, 30, 6, 1, 127, 357, 441, 315, 140, 42, 7, 1, 323, 1016, 1428, 1176, 630, 224, 56, 8, 1, 835, 2907, 4572, 4284, 2646, 1134, 336, 72, 9, 1, 2188, 8350, 14535, 15240, 10710, 5292, 1890, 480, 90, 10, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS Number of ordered trees with n edges having k leaves at even height. Row sums are the Catalan numbers (A000108). T(n,0)=A001006(n-1) (the Motzkin numbers). Sum_{k=0..n-1} k*T(n,k) = binomial(2n-2, n-2) = A001791(n-1). Mirror image of A091187. T(n,k) is the number of Dyck paths of semilength n and having k dud's (here u=(1,1) and d=(1,-1)). Example: T(4,2)=3 because we have uud(du[d)ud], uu(dud)(dud) and uu(du[d)ud]d (the dud's are shown between parentheses). T(n,k) is the number of Dyck paths of semilength n and containing exactly k double rises whose matching down steps form a doublefall. Example: UUUDUDDD has 2 double rises but only the first has matching Ds - the path's last 2 steps - forming a doublefall. (Travel horizontally east from an up step to encounter its matching down step.) - David Callan, Jul 15 2004 T(n,k) is the number of ordered trees on n edges containing k edges of outdegree 1. (The outdegree of an edge is the outdegree of its child vertex. Thus edges of outdegree 1 correspond to non-root vertices of outdegree 1.) T(3,2)=2 because   /\.../\.   |.....|. each have one edge of outdegree 1. - David Callan, Oct 25 2004 Exponential Riordan array [exp(x)*Bessel_I(1,2x)/x, x]. - Paul Barry, Mar 09 2010 REFERENCES J. L. Baril, S. Kirgizov, The pure descent statistic on permutations, Preprint, 2016, http://jl.baril.u-bourgogne.fr/Stirling.pdf. See Table 2. LINKS Alois P. Heinz, Rows n = 1..200, flattened David Callan, Bijections for Dyck paths with all peak heights of the same parity, arXiv:1702.06150 [math.CO], 2017. M. Dziemianczuk, Enumerations of plane trees with multiple edges and Raney lattice paths, Discrete Mathematics 337 (2014): 9-24. A. Sapounakis, I. Tasoulas and P. Tsikouras, Counting strings in Dyck paths, Discrete Math., 307 (2007), 2909-2924. Yidong Sun, The statistic "number of udu's" in Dyck paths, Discrete Math., 287 (2004), 177-186. FORMULA T(n, k) = binomial(n-1, k)*(Sum_{j=0..ceiling((n-k)/2)} binomial(n-k, j)*binomial(n-k-j, j-1))/(n-k) for 0 <= k < n; T(n, k)=0 for k >= n. G.f.: G = G(t, z) satisfies z*G^2 - (1 + z - t*z)*G + 1 + z - t*z = 0. T(n, k) = M(n-k-1)*binomial(n-1, k), where M(n) = A001006(n) are the Motzkin numbers. T(n+1, k+1) = n*T(n, k)/(k+1). - David Callan, Dec 09 2004 G.f.: 1/(1-x-xy-x^2/(1-x-xy-x^2/(1-x-xy-x^2/(1-x-xy-x^2/(1-... (continued fraction). - Paul Barry, Aug 03 2009 E.g.f.: exp(x+xy)*Bessel_I(1,2x)/x. - Paul Barry, Mar 10 2010 EXAMPLE T(4,1)=6 because we have u(ud)dudud, udu(ud)dud, ududu(ud)d, uuudd(ud)d, u(ud)uuddd and uuu(ud)ddd (here u=(1,1), d=(1,-1) and the peaks at even height are shown between parentheses). Triangle begins:    1;    1,   1;    2,   2,   1;    4,   6,   3,  1;    9,  16,  12,  4,  1;   21,  45,  40, 20,  5, 1;   51, 126, 135, 80, 30, 6, 1; MAPLE T := proc(n, k) if k0, b(x-1, y-1, 0)*z^irem(t*y+t, 2), 0)+       `if`(y (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0\$2)): seq(T(n), n=1..16);  # Alois P. Heinz, May 12 2017 MATHEMATICA (* m = MotzkinNumber *) m = 1; m[n_] := m[n] = m[n - 1] + Sum[m[k]*m[n - 2 - k], {k, 0, n - 2}]; t[n_, n_] = 1; t[n_, k_] := m[n - k]*Binomial[n - 1, k - 1]; Table[t[n, k], {n, 1, 11}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 10 2013 *) CROSSREFS Cf. A000108, A001006, A001791, A091187, A243752. Sequence in context: A068957 A119468 A175136 * A112307 A228336 A111062 Adjacent sequences:  A091866 A091867 A091868 * A091870 A091871 A091872 KEYWORD nonn,tabl AUTHOR Emeric Deutsch, Mar 10 2004 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified October 15 09:22 EDT 2019. Contains 328026 sequences. (Running on oeis4.)