OFFSET
0,3
COMMENTS
Hankel transform is [1,1,1,...] = A000012. - Paul Barry, Jan 19 2009
The inverse Motzkin transform apparently yields 1 followed by A000930, which implies a generating function g(x)=1+z/(1-z-z^3) where z=x*A001006(x). - R. J. Mathar, Jul 07 2009
It appears that the infinite set of interpolated sequences between the Motzkin and the Catalan can be generated with a succession of INVERT transforms, given each sequence has two leading 1's. Also, the N-th sequence in the set starting with (N=1, A001006) can be generated from a production matrix of the form "M" in the formula section, such that the main diagonal is (N leading 1's, 0, 0, 0, ...). M with a diagonal of (1, 0, 0, 0, ...) generates A001006, while M with a main diagonal of all 1's is the production matrix for A000108. - Gary W. Adamson, Jul 29 2011
From Gus Wiseman, Jun 22 2019: (Start)
Conjecture: Also the number of non-capturing set partitions of {1..n} (A326254). A set partition is capturing if it has two blocks of the form {...x...y...} and {...z...t...} where x < z and y > t or x > z and y < t. This is a weaker condition than nesting, so for example {{1,3,5},{2,4}} is capturing but not nesting. The a(0) = 1 through a(4) = 14 non-capturing set partitions are:
{} {{1}} {{1,2}} {{1,2,3}} {{1,2,3,4}}
{{1},{2}} {{1},{2,3}} {{1},{2,3,4}}
{{1,2},{3}} {{1,2},{3,4}}
{{1,3},{2}} {{1,2,3},{4}}
{{1},{2},{3}} {{1,2,4},{3}}
{{1,3},{2,4}}
{{1,3,4},{2}}
{{1},{2},{3,4}}
{{1},{2,3},{4}}
{{1,2},{3},{4}}
{{1},{2,4},{3}}
{{1,3},{2},{4}}
{{1,4},{2},{3}}
{{1},{2},{3},{4}}
(End)
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
E. Barcucci et al., From Motzkin to Catalan Permutations, Discr. Math., 217 (2000), 33-49.
Jean-Luc Baril and Sergey Kirgizov, Bijections from Dyck and Motzkin meanders with catastrophes to pattern avoiding Dyck paths, arXiv:2104.01186 [math.CO], 2021.
Paul Barry, On sequences with {-1, 0, 1} Hankel transforms, arXiv preprint arXiv:1205.2565 [math.CO], 2012. - From N. J. A. Sloane, Oct 18 2012
Petr Gregor, Torsten Mütze, and Namrata, Combinatorial generation via permutation languages. VI. Binary trees, arXiv:2306.08420 [cs.DM], 2023.
Nickolas Hein and Jia Huang, Variations of the Catalan numbers from some nonassociative binary operations, arXiv:1807.04623 [math.CO], 2018.
J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5
Toufik Mansour and Mark Shattuck, Pattern Avoiding Partitions, Sequence A054391 and the Kernel Method, Applications and Applied Mathematics, Vol. 6, Issue 2 (December 2011), pp. 397-411.
T. Mansour and M. Shattuck, Restricted partitions and generalized Catalan numbers, PU. M. A., Vol. (2011), No. 2, pp. 239-251. - From N. J. A. Sloane, Oct 13 2012
Eric Marberg, Crossings and nestings in colored set partitions, arXiv preprint arXiv:1203.5738 [math.CO], 2012.
FORMULA
G.f.: 1 - 2*x^2 / (2*x^2 - 3*x + 1 - sqrt(1 - 2*x - 3*x^2)). - Mansour and Shattuck
G.f.: 1/(1-x-x^2/(1-2x-x^2/(1-x-x^2/(1-x-x^2/(1-x-x^2/(1-x-x^2/(1-... (continued fraction) (conjecture). - Paul Barry, Jan 19 2009
From Gary W. Adamson, Jul 29 2011: (start)
a(n) = upper left term of M^n, a(n+1) = sum of top row terms of M^n; M = an infinite square production matrix as follows with a main diagonal of (1, 1, 1, 0, 0, 0, ...):
1, 1, 0, 0, 0, 0, ...
1, 1, 1, 0, 0, 0, ...
1, 1, 1, 1, 0, 0, ...
1, 1, 1, 0, 1, 0, ...
1, 1, 1, 1, 0, 1, ...
1, 1, 1, 1, 1, 0, ...
... (End)
a(n) = Sum_{k=1..n-1} (sum(l=1..k, (binomial(k,l)*l*sum(j=0..n+l-k-1, binomial(j,1-n-2*l+k+2*j)*binomial(n-1+l-k,j)))/(n+l-k-1))) + 1. - Vladimir Kruchinin, Oct 31 2011
D-finite with recurrence (-n+1)*a(n) + 3*(2*n-3)*a(n-1) + (-8*n+11)*a(n-2) + (-5*n+32)*a(n-3) + (7*n-31)*a(n-4) + 3*(-n+4)*a(n-5)= 0. - R. J. Mathar, Nov 26 2012
G.f.: 1 - x*(2*x^2-3*x+1 + 1/G(0))/(2*(x^3-3*x^2+4*x-1)), where G(k)= 1 + x*(2+3*x)*(4*k+1)/( 4*k+2 - x*(2+3*x)*(4*k+2)*(4*k+3)/(x*(2+3*x)*(4*k+3) + 4*(k+1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jun 29 2013
EXAMPLE
a(4) = 14, a(5) = 41 since the top row of M^4 = (14, 14, 9, 3, 1), with 41 = (14 + 14 + 9 + 3 + 1).
MAPLE
c := x->(1-sqrt(1-4*x))/(2*x); a := (x, j)->(x)/((1-4*x)*(c(x))^2*(1-c(x))^(j))*(-x^2*(c(x))^2*(1-c(x))*(x^2*(c(x))^4)^(j)-(1-3*x-2*x^2)*(c(x))^2*(x*(c(x))^2)^(j)+x);
b := (x, j)->1+(1)/((1-4*x)*c(x)*(1-c(x))^(j))*(-2*x^3*(c(x))^2*(x^2*(c(x))^4)^(j)+(1-3*x-2*x^2)*c(x)*(x*(c(x))^2)^(j)-2*x^2);
co := (x, j)->(1)/((1-4*x)*(1-c(x))^(j))*(x^2*(x^2*(c(x))^4)^(j)-(1-3*x-2*x^2)*(x*(c(x))^2)^(j)+x^2);
MATHEMATICA
CoefficientList[Series[1 - 2*x^2/(2*x^2 - 3*x + 1 - Sqrt[1 - 2*x - 3*x^2]), {x, 0, 50}], x] (* G. C. Greubel, Apr 27 2017 *)
PROG
(Maxima) a(n):=sum((sum((binomial(k, l)*l*sum(binomial(j, 1-n-2*l+k+2*j)*binomial(n-1+l-k, j), j, 0, n+l-k-1))/(n+l-k-1), l, 1, k)), k, 1, n-1)+1; \\ Vladimir Kruchinin, Oct 31 2011
(PARI) x='x+O('x^66); gf=1-2*x^2/(2*x^2-3*x+1-sqrt(1-2*x-3*x^2)); Vec(gf) \\ Joerg Arndt, Jun 29 2013
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Elisa Pergola (elisa(AT)dsi.unifi.it), May 21 2000
STATUS
approved