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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.

%I A281260 #51 Sep 28 2023 13:02:01
%S A281260 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,
%T A281260 2,48,280,560,420,112,8,0,2,63,504,1470,1764,882,168,9,0,2,80,840,
%U A281260 3360,5880,4704,1680,240,10,0,2,99,1320,6930,16632,19404,11088,2970,330,11,0,2,120,1980,13200,41580
%N 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.
%C A281260 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.
%C A281260 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.
%C A281260 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).
%C A281260 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.
%H A281260 Michael De Vlieger, <a href="/A281260/b281260.txt">Table of n, a(n) for n = 1..11325</a> (rows n = 1..150, flattened)
%H A281260 David Callan, <a href="/A281260/a281260.pdf">Generalized Narayana Numbers</a>
%H A281260 Vladimir Kruchinin, Dmitry Kruchinin, and Yuriy Shablya, <a href="http://kpmit.dvfu.ru/conf2019/abstracts/Kruchinin/Abstract_TUSUR.pdf">On some properties of generalized Narayana numbers</a>, Tomsk State University of Control Systems and Radioelectronics, (Tomsk, Russia 2019).
%H A281260 Feiyang Lin, <a href="http://www-users.math.umn.edu/~reiner/REU/REU2020notes/Problem3_REUreport.pdf">F-polynomials for the R-Kronecker quiver</a>, University of Minnesota, Research Experiences for Undergrads (2020).
%H A281260 Bo Wang and Candice X.T. Zhang, <a href="https://arxiv.org/abs/2309.05903">Interlacing property of a family of generating polynomials over Dyck paths</a>, arXiv:2309.05903 [math.CO], 2023.
%H A281260 Yi Wang and Arthur L.B. Yang, <a href="https://arxiv.org/abs/1702.07822">Total positivity of Narayana matrices</a>, arXiv:1702.07822 [math.CO], 2017.
%H A281260 James J. Y. Zhao, <a href="https://arxiv.org/abs/2108.03590">On the positive zeros of generalized Narayana polynomials related to the Boros-Moll polynomials</a>, arXiv:2108.03590 [math.CO], 2021.
%F A281260 Row sums are A033184(n+1,2).
%F A281260 The same triangle as A108838 with reversed rows but without leading column.
%F A281260 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
%F A281260 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
%e A281260 The triangle begins:
%e A281260 n\k:  0  1    2     3      4      5      6      7      8     9   10  11  . . .
%e A281260 01 :  1
%e A281260 02 :  0  2
%e A281260 03 :  0  2    3
%e A281260 04 :  0  2    8     4
%e A281260 05 :  0  2   15    20      5
%e A281260 06 :  0  2   24    60     40      6
%e A281260 07 :  0  2   35   140    175     70      7
%e A281260 08 :  0  2   48   280    560    420    112      8
%e A281260 09 :  0  2   63   504   1470   1764    882    168      9
%e A281260 10 :  0  2   80   840   3360   5880   4704   1680    240    10
%e A281260 11 :  0  2   99  1320   6930  16632  19404  11088   2970   330   11
%e A281260 12 :  0  2  120  1980  13200  41580  66528  55440  23760  4950  440  12
%e A281260 etc.
%t A281260 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 *)
%Y A281260 Cf. A000007, A001263, A033184, A088218, A103365, A108838, A145596, A281293, A281297.
%K A281260 nonn,tabl,easy
%O A281260 1,3
%A A281260 _Werner Schulte_, Jan 18 2017