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A281260
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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.
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4
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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
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OFFSET
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1,3
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COMMENTS
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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.
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LINKS
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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).
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.
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FORMULA
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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
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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CROSSREFS
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Cf. A000007, A001263, A033184, A088218, A103365, A108838, A145596, A281293, A281297.
Sequence in context: A265583 A339754 A238156 * A182406 A160706 A087509
Adjacent sequences: A281257 A281258 A281259 * A281261 A281262 A281263
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KEYWORD
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nonn,tabl,easy
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AUTHOR
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Werner Schulte, Jan 18 2017
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STATUS
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approved
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