|
%I #51 Sep 08 2022 08:45:31
%S 1,1,2,1,5,1,9,5,1,14,21,1,20,56,14,1,27,120,84,1,35,225,300,42,1,44,
%T 385,825,330,1,54,616,1925,1485,132,1,65,936,4004,5005,1287,1,77,1365,
%U 7644,14014,7007,429
%N Triangle (read by rows) with row sums = Motzkin sums (also called Riordan numbers) (A005043): T(n,s) = (1/n)*C(n,s)*(C(n-s,s+1) - C(n-s-2,s-1)).
%C Whereas A005043 counts certain trees, or noncrossed partitions, this subdivides the counts according to the number of leaves, or the lattice rank. Analogous to the Narayana triangle (A001263), where rows sum to the Catalan numbers.
%C Diagonals of A132081 are rows of A033282. - _Tom Copeland_, May 08 2012
%C Related to the number of certain non-crossing partitions for the root system A_n. Cf. p. 12, Athanasiadis and Savvidou. See also A108263 and A100754. - _Tom Copeland_, Oct 19 2014
%H C. Athanasiadis and C. Savvidou, <a href="http://arxiv.org/abs/1204.0362">The local h-vector of the cluster subdivision of a simplex</a>, arXiv preprint arXiv:1204.0362 [math.CO], 2012.
%H F. R. Bernhart, <a href="http://dx.doi.org/10.1016/S0012-365X(99)00054-0">Catalan, Motzkin and Riordan numbers</a>, Discr. Math., 204 (1999), 73-112.
%H F. R. Bernhart & N. J. A. Sloane, <a href="/A006343/a006343.pdf">Emails, April-May 1994</a>
%H T. Copeland, <a href="http://tcjpn.wordpress.com/2015/12/21/generators-inversion-and-matrix-binomial-and-integral-transforms/">Generators, Inversion, and Matrix, Binomial, and Integral Transforms</a>, 2015.
%F a(n,k) = binomial(n,k)*binomial(n-2-k,k)/(k+1). - _David Callan_, Jul 22 2008
%F From _Peter Bala_, Oct 22 2008: (Start)
%F O.g.f. : 1 + x + sqrt(1 - 2*x + x^2*(1 - 4*a))]/(2*x*(1 + a*x)) = 1 + a*x^2 + a*x^3 + (a + 2*a^2)*x^4 + (a + 5*a^2)*x^5 + (a + 9*a^2 + 5*a^3)*x^6 + ... .
%F Define a functional I on formal power series of the form f(x) = 1 + a*x + b*x^2 + ... by the following iterative process. Define inductively f^(1)(x) = f(x) and f^(n+1)(x) = f(x*f^(n)(x)) for n >= 1. Then set I(f(x)) = lim n -> infinity f^(n)(x) in the x-adic topology on the ring of formal power series; the operator I may also be defined by I(f(x)) := 1/x*series reversion of x/f(x).
%F Let now f(x) = 1 + a*x^2 + a*x^3 + a*x^4 + ... . Then the o.g.f. for this table is I(f(x)) = 1 + a*x^2 + a*x^3 + (a + 2*a^2)*x^4 + (a + 5*a^2)*x^5 + (a + 9*a^2 + 5*a^3)*x^6 + ... . Cf. A001263 and A108767. (End)
%e A005043(6) = 15 = 1+9+5 since NC (noncrossed, planar) partitions of 6-point cycle without singletons have 1,9,5 items with 1,2,3 blocks.
%e Triangle begins:
%e 1;
%e 1, 2;
%e 1, 5;
%e 1, 9, 5;
%e 1, 14, 21;
%e 1, 20, 56, 14;
%e 1, 27, 120, 84;
%e 1, 35, 225, 300, 42;
%e 1, 44, 385, 825, 330;
%e ...
%t Map[Most, Table[(1/n) Binomial[n, s] (Binomial[n - s, s + 1] - Binomial[n - s - 2, s - 1]), {n, 3, 14}, {s, 0, n}] /. k_ /; k <= 0 -> Nothing] // Flatten (* _Michael De Vlieger_, Jan 09 2016 *)
%o (Magma) /* triangle excluding 0 */ [[Binomial(n,k)*Binomial(n-2-k,k)/(k+1): k in [0..n-3]]: n in [3..15]]; // _Vincenzo Librandi_, Oct 19 2014
%Y Row sums are A007404.
%Y Cf. A001263, A005043, A033282, A100754, A108263, A108767, A132081.
%K nonn,tabf
%O 3,3
%A Frank R. Bernhart (farb45(AT)gmail.com), Oct 30 2007
%E Edited by _N. J. A. Sloane_, Jul 01 2008 at the suggestion of _R. J. Mathar_
%E Name corrected by _Emeric Deutsch_, Dec 20 2014
|
