A243662 Triangle read by rows: the reversed x = 1+q Narayana triangle at m=2.

1, 3, 1, 12, 8, 1, 55, 55, 15, 1, 273, 364, 156, 24, 1, 1428, 2380, 1400, 350, 35, 1, 7752, 15504, 11628, 4080, 680, 48, 1, 43263, 100947, 92169, 41895, 9975, 1197, 63, 1, 246675, 657800, 708400, 396704, 123970, 21560, 1960, 80, 1, 1430715, 4292145, 5328180, 3552120, 1381380, 318780, 42504, 3036, 99, 1

Offset: 1

Author

N. J. A. Sloane, Jun 13 2014

Keywords

• nonn
• tabl

Comments

See Novelli-Thibon (2014) for precise definition.

From Tom Copeland, Dec 13 2022: (Start)

The row polynomials are the nonvanishing numerator polynomials generated in the compositional, or Lagrange, inversion in x about the origin of the odd o.g.f. Od1(x,t) = x*(t*(1-x^2)-x^2) / (1-x^2) = t*x - x^3 - x^5 - x^7 - x^9 - ... .

For example, from the Lagrange inversion formula (LIF), the tenth derivative in x of (x/Od1(x,t))^11 / 11! = (1/((t*(1-x^2)-x^2) / (1-x^2)))^11 / 11! at x = 0 is (t^4 + 24*t^3 + 156*t^2 + 364*t + 273) / t^16. These polynomials are also generated by the iterated derivatives ((1/(D Od1(x,t)) D)^n g(x) evaluated at x = 0 where D = d/dx.

An explicit generating function for the polynomials can be obtained by finding the solution of the cubic equation y - t*x - y*x^2 + (1+t)*x^3 = 0 for x in terms of y and t that satisfies y(x=0;t) = 0 = x(y=0;t).

The row polynomials are also the polynomials generated in the compositional inverse of O(x,t) = x / (1+(1+t)x)*(1+x)^2) = x + (-t - 3)*x^2 + (t^2 + 4 t + 6)*x^3 + (-t^3 - 5*t^2 - 10*t - 10)*x^4 + ..., containing the truncated Pascal polynomials of A104712 / A325000.

For example, from the LIF, the third derivative of ((1 + (1+t)*x)*(1+x)^2)^4 / 4! at x = 0 is 55 + 55*t + 15*t^2 + t^3.

A natural refinement of this array was provided in a letter by Isaac Newton in 1676--a set of partition polynomials for generating the o.g.f. of the compositional inverse of the generic odd o.g.f. x + u_1 x^3 + u_2 x^5 + ... in the infinite set of indeterminates u_n. (End)

T(n,k) is the number of noncrossing cacti with n+1 nodes and n+1-k blocks. See A361242. - Andrew Howroyd, Apr 13 2023

Examples

			Triangle begins:
     1;
     3,    1;
    12,    8,    1;
    55,   55,   15,   1;
   273,  364,  156,  24,  1;
  1428, 2380, 1400, 350, 35, 1;
  ...
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows 1 <= n <= 150, flattened)
• Paul Barry, On the inversion of Riordan arrays, arXiv:2101.06713 [math.CO], 2021.
• J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014. See Fig. 10.

Crossrefs

Cf. A001764, A001263, A243663 (m=3).

Row sums give A003168.

Row reversed triangle is A102537.

Programs

Mathematica

T[m_][n_, k_] := Binomial[(m + 1) n + 1 - k, n - k] Binomial[n, k - 1]/n;
Table[T[2][n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Feb 12 2019 *)

PARI

T(n)=[Vecrev(p) | p<-Vec(serreverse(x/((1+x+x*y)*(1+x)^2) + O(x*x^n)))]
{ my(A=T(10)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Apr 13 2023

Formula

T(n,k) = (binomial(3*n+1,n) * binomial(n,k-1) * binomial(n-1,k-1)) / (binomial(3*n,k-1) * (3*n+1)) = (A001764(n) * A001263(n,k) * k) / binomial(3*n,k-1) for 1 <= k <= n (conjectured). - Werner Schulte, Nov 22 2018

T(n,k) = binomial(3*n+1-k,n-k) * binomial(n,k-1) / n for 1 <= k <= n, more generally: T_m(n,k) = binomial((m+1)*n+1-k,n-k) * binomial(n,k-1) / n for 1 <= k <= n and some fixed integer m > 1. - Werner Schulte, Nov 22 2018

G.f.: A(x,y) is the series reversion of x/((1 + x + x*y)*(1 + x)^2). - Andrew Howroyd, Apr 13 2023

Extensions

Data and Example (T(2,2) and T(5,3)) corrected and more terms added by Werner Schulte, Nov 22 2018