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A132081 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)).

%I A132081 #54 Sep 22 2024 17:49:45
%S A132081 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 A132081 385,825,330,1,54,616,1925,1485,132,1,65,936,4004,5005,1287,1,77,1365,
%U A132081 7644,14014,7007,429
%N A132081 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 A132081 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 A132081 Diagonals of A132081 are rows of A033282. - _Tom Copeland_, May 08 2012
%C A132081 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 A132081 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 A132081 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 A132081 F. R. Bernhart & N. J. A. Sloane, <a href="/A006343/a006343.pdf">Emails, April-May 1994</a>
%H A132081 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 A132081 a(n,k) = binomial(n,k)*binomial(n-2-k,k)/(k+1). - _David Callan_, Jul 22 2008
%F A132081 From _Peter Bala_, Oct 22 2008: (Start)
%F A132081 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 + ... . [corrected by _Jason Yuen_, Sep 22 2024]
%F A132081 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 A132081 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 A132081 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 A132081 Triangle begins:
%e A132081   1;
%e A132081   1,   2;
%e A132081   1,   5;
%e A132081   1,   9,   5;
%e A132081   1,  14,  21;
%e A132081   1,  20,  56,  14;
%e A132081   1,  27, 120,  84;
%e A132081   1,  35, 225, 300,  42;
%e A132081   1,  44, 385, 825, 330;
%e A132081   ...
%t A132081 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 A132081 (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 A132081 Row sums are A007404.
%Y A132081 Cf. A001263, A005043, A033282, A100754, A108263, A108767, A132081.
%K A132081 nonn,tabf
%O A132081 3,3
%A A132081 Frank R. Bernhart (farb45(AT)gmail.com), Oct 30 2007
%E A132081 Edited by _N. J. A. Sloane_, Jul 01 2008 at the suggestion of _R. J. Mathar_
%E A132081 Name corrected by _Emeric Deutsch_, Dec 20 2014