A281260 Triangular array of generalized Narayana numbers T(n,k) = 2*binomial(n+1,k)* binomial(n-2,k-1)/(n+1) for n >= 1 and 0 <= k <= n-1, read by rows.

1, 0, 2, 0, 2, 3, 0, 2, 8, 4, 0, 2, 15, 20, 5, 0, 2, 24, 60, 40, 6, 0, 2, 35, 140, 175, 70, 7, 0, 2, 48, 280, 560, 420, 112, 8, 0, 2, 63, 504, 1470, 1764, 882, 168, 9, 0, 2, 80, 840, 3360, 5880, 4704, 1680, 240, 10, 0, 2, 99, 1320, 6930, 16632, 19404, 11088, 2970, 330, 11, 0, 2, 120, 1980, 13200, 41580

Offset: 1

Author

Werner Schulte, Jan 18 2017

Keywords

• nonn
• tabl
• easy

Comments

The current array is the case m = 1 of the generalized Narayana numbers N_m(n,k) := (m+1)/(n+1)*binomial(n+1,k)*binomial(n-m-1,k-1) for m >= 0, n >= m and 0 <= k <= n-m with N_m(n,0) = A000007(n-m). Case m = 0 gives the table of Narayana numbers A001263 without leading column N_0(n,0) = A000007(n). For m = 2 see A281293, and for m = 3 see A281297. For combinatorial interpretations see the link to: David Callan, Generalized Narayana Numbers.

The polynomials p(m,n,x) = Sum_{k=0..n-m} N_m(n,k)*x^k satisfy the recurrence equation: x*p"(m,n,x) = n*(n-m-1)*p(m,n-1,x) for n > m >= 0 (p" is the second derivative of p), i.e.: k*(k-1)*N_m(n,k) = n*(n-m-1)*N_m(n-1,k-1) for k > 0 and n > m >= 0. Furthermore: Sum_{k=0..n-m} binomial(n+1-k,m+1)*N_m(n,k) = binomial(n,m)*A088218(n-m) for n >= m >= 0.

There is a relationship of these N_m(n,k) to those N_r(n,k) of A145596 (see the second comment): N_m(n+1,k) = N_r(n,k)*binomial(k+r,r)/binomial(n,r) for k >= 1 and 1 <= m = r <= n, and alternatively: N_r(n,k) = N_m(n+1,k)*binomial(n,m)/ binomial(k+m,m).

Conjecture: Sum_{k=1..n-m} binomial(n+1-k,m) * N_m(n,k) * A103365(n+1-m-k) = (m+1)^2 * A000007(n-m-1) for n > m >= 0.

Examples

			The triangle begins:
n\k:  0  1    2     3      4      5      6      7      8     9   10  11  . . .
01 :  1
02 :  0  2
03 :  0  2    3
04 :  0  2    8     4
05 :  0  2   15    20      5
06 :  0  2   24    60     40      6
07 :  0  2   35   140    175     70      7
08 :  0  2   48   280    560    420    112      8
09 :  0  2   63   504   1470   1764    882    168      9
10 :  0  2   80   840   3360   5880   4704   1680    240    10
11 :  0  2   99  1320   6930  16632  19404  11088   2970   330   11
12 :  0  2  120  1980  13200  41580  66528  55440  23760  4950  440  12
etc.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows n = 1..150, flattened)
• David Callan, Generalized Narayana Numbers
• Vladimir Kruchinin, Dmitry Kruchinin, and Yuriy Shablya, On some properties of generalized Narayana numbers, Tomsk State University of Control Systems and Radioelectronics, (Tomsk, Russia 2019).
• Feiyang Lin, F-polynomials for the R-Kronecker quiver, University of Minnesota, Research Experiences for Undergrads (2020).
• Bo Wang and Candice X.T. Zhang, Interlacing property of a family of generating polynomials over Dyck paths, arXiv:2309.05903 [math.CO], 2023.
• Yi Wang and Arthur L.B. Yang, Total positivity of Narayana matrices, arXiv:1702.07822 [math.CO], 2017.
• James J. Y. Zhao, On the positive zeros of generalized Narayana polynomials related to the Boros-Moll polynomials, arXiv:2108.03590 [math.CO], 2021.

Crossrefs

Cf. A000007, A001263, A033184, A088218, A103365, A108838, A145596, A281293, A281297.

Programs

Mathematica

Table[2 Binomial[n + 1, k] Binomial[n - 2, k - 1]/(n + 1), {n, 1, 12}, {k, 0, n - 1}] // Flatten (* Michael De Vlieger, Jan 19 2017 *)

Formula

Row sums are A033184(n+1,2).

The same triangle as A108838 with reversed rows but without leading column.

G.f.: ((x*y-x-1)*sqrt(x^2*y^2+(-2*x^2-2*x)*y+x^2-2*x+1)+x^2*y^2+(-2*x^2-2*x)*y+x^2+1)/(2*x). - Vladimir Kruchinin, Oct 11 2020

G.f. satisfies x*A(x,y)^2-(x^2*y^2+((-2)*x^2-2*x)*y+x^2+1)*A(x,y)+x=0. - Vladimir Kruchinin, Oct 11 2020