The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 Please make a donation to keep the OEIS running. We are now in our 56th year. In the past year we added 10000 new sequences and reached almost 9000 citations (which often say "discovered thanks to the OEIS"). Other ways to donate

 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 T(n, k) is the number of Dyck paths of semilength n and having k udu's (here u=(1,1) and d=(1,-1)). Note that reversing a path swaps u and d, thus udu becomes dud and vice versa. - Michael Somos, Feb 26 2020 LINKS Alois P. Heinz, Rows n = 1..200, flattened J. L. Baril and S. Kirgizov, The pure descent statistic on permutations, Preprint, 2016. See Table 2. 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[0] = 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 *) PROG (PARI) {T(n, k) = my(y, c, w); if( k<0 || k>=n, 0, w = vector(n);   forvec(v=vector(2*n, k, [0, 1]), c=y=0; for(k=1, 2*n, if( 0>(y += (-1)^v[k]), break)); if( y, next); for(i=1, 2*n-2, c += ([0, 1, 0] == v[i..i+2])); w[c+1]++); w[k+1])}; /* Michael Somos, Feb 26 2020 */ 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 December 1 05:26 EST 2020. Contains 338833 sequences. (Running on oeis4.)