A281293 -id:A281293 - cp's OEIS frontend

A281260 Triangular array of generalized Narayana numbers T(n,k) = 2binomial(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

Werner Schulte, Jan 18 2017

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.

			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.

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.

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

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 *)

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

A281297 Triangular array of generalized Narayana numbers T(n,k) = 4binomial(n+1,k) binomial(n-4,k-1)/(n+1) for n >= 3 and 0 <= k <= n-3, read by rows.

Original entry on oeis.org

1, 0, 4, 0, 4, 10, 0, 4, 24, 20, 0, 4, 42, 84, 35, 0, 4, 64, 224, 224, 56, 0, 4, 90, 480, 840, 504, 84, 0, 4, 120, 900, 2400, 2520, 1008, 120, 0, 4, 154, 1540, 5775, 9240, 6468, 1848, 165, 0, 4, 192, 2464, 12320, 27720, 29568, 14784, 3168, 220, 0, 4, 234, 3744, 24024, 72072, 108108, 82368, 30888, 5148

Offset: 3

Werner Schulte, Jan 19 2017

The current array is the case m = 3 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 = 1 see A281260, and for m = 2 see A281293.

			The triangle begins:
n\k:  0  1    2     3      4      5       6      7      8     9   10  . . .
03 :  1
04 :  0  4
05 :  0  4   10
06 :  0  4   24    20
07 :  0  4   42    84     35
08 :  0  4   64   224    224     56
09 :  0  4   90   480    840    504      84
10 :  0  4  120   900   2400   2520    1008    120
11 :  0  4  154  1540   5775   9240    6468   1848    165
12 :  0  4  192  2464  12320  27720   29568  14784   3168   220
13 :  0  4  234  3744  24024  72072  108108  82368  30888  5148  286
etc.

David Callan, Generalized Narayana Numbers .

Cf. A000007, A001263, A033184, A281260, A281293.

Row sums are A033184(n+1,4).

G.f.: A(x) = x*A281260(x,y)^2. - Vladimir Kruchinin, Oct 10 2020

cp's OEIS Frontend

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

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A281297 Triangular array of generalized Narayana numbers T(n,k) = 4binomial(n+1,k) binomial(n-4,k-1)/(n+1) for n >= 3 and 0 <= k <= n-3, read by rows.

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cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A281297 Triangular array of generalized Narayana numbers T(n,k) = 4*binomial(n+1,k)* binomial(n-4,k-1)/(n+1) for n >= 3 and 0 <= k <= n-3, read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

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

A281297 Triangular array of generalized Narayana numbers T(n,k) = 4binomial(n+1,k) binomial(n-4,k-1)/(n+1) for n >= 3 and 0 <= k <= n-3, read by rows.