%I #51 Sep 28 2023 13:02:01
%S 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 2,48,280,560,420,112,8,0,2,63,504,1470,1764,882,168,9,0,2,80,840,
%U 3360,5880,4704,1680,240,10,0,2,99,1320,6930,16632,19404,11088,2970,330,11,0,2,120,1980,13200,41580
%N 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 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 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 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 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 Michael De Vlieger, <a href="/A281260/b281260.txt">Table of n, a(n) for n = 1..11325</a> (rows n = 1..150, flattened)
%H David Callan, <a href="/A281260/a281260.pdf">Generalized Narayana Numbers</a>
%H 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 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 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 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 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 Row sums are A033184(n+1,2).
%F The same triangle as A108838 with reversed rows but without leading column.
%F 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 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 The triangle begins:
%e n\k: 0 1 2 3 4 5 6 7 8 9 10 11 . . .
%e 01 : 1
%e 02 : 0 2
%e 03 : 0 2 3
%e 04 : 0 2 8 4
%e 05 : 0 2 15 20 5
%e 06 : 0 2 24 60 40 6
%e 07 : 0 2 35 140 175 70 7
%e 08 : 0 2 48 280 560 420 112 8
%e 09 : 0 2 63 504 1470 1764 882 168 9
%e 10 : 0 2 80 840 3360 5880 4704 1680 240 10
%e 11 : 0 2 99 1320 6930 16632 19404 11088 2970 330 11
%e 12 : 0 2 120 1980 13200 41580 66528 55440 23760 4950 440 12
%e etc.
%t 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 Cf. A000007, A001263, A033184, A088218, A103365, A108838, A145596, A281293, A281297.
%K nonn,tabl,easy
%O 1,3
%A _Werner Schulte_, Jan 18 2017