A281260 -id:A281260 - cp's OEIS frontend

A132813 Triangle read by rows: A001263 * A127648 as infinite lower triangular matrices.

1, 1, 2, 1, 6, 3, 1, 12, 18, 4, 1, 20, 60, 40, 5, 1, 30, 150, 200, 75, 6, 1, 42, 315, 700, 525, 126, 7, 1, 56, 588, 1960, 2450, 1176, 196, 8, 1, 72, 1008, 4704, 8820, 7056, 2352, 288, 9, 1, 90, 1620, 10080, 26460, 31752, 17640, 4320, 405, 10

Offset: 0

Gary W. Adamson, Sep 01 2007

Also T(n,k) = binomial(n-1, k-1)*binomial(n, k-1), related to Narayana polynomials (see Sulanke reference). - Roger L. Bagula, Apr 09 2008

h-vector for cluster complex associated to the root system B_n. See p. 8, Athanasiadis and C. Savvidou. - Tom Copeland, Oct 19 2014

			First few rows of the triangle are:
  1;
  1,  2;
  1,  6,   3;
  1, 12,  18,   4;
  1, 20,  60,  40,   5;
  1, 30, 150, 200,  75,   6;
  1, 42, 315, 700, 525, 126, 7;
  ...

Reinhard Zumkeller, Rows n = 0..125 of table, flattened
N. Alexeev and A. Tikhomirov, Singular Values Distribution of Squares of Elliptic Random Matrices and type-B Narayana Polynomials, arXiv preprint arXiv:1501.04615 [math.PR], 2015.
C. Athanasiadis and C. Savvidou, The local h-vector of the cluster subdivision of a simplex, arXiv preprint arXiv:1204.0362 [math.CO], 2012.
Robert. A. Sulanke, Counting Lattice Paths by Narayana Polynomials Electronic J. Combinatorics 7, No. 1, R40, 1-9, 2000.

Family of polynomials (see A062145): A008459 (c=1), this sequence (c=2), A062196 (c=3), A062145 (c=4), A062264 (c=5), A062190 (c=6).
Columns: A000012 (k=0), A002378 (k=1), A006011 (k=2), 4*A006542 (k=3), 5*A006857 (k=4), 6*A108679 (k=5), 7*A134288 (k=6), 8*A134289 (k=7), 9*A134290 (k=8), 10*A134291 (k=9).
Diagonals: A000027 (k=n), A002411 (k=n-1), A004302 (k=n-2), A108647 (k=n-3), A134287 (k=n-4).
Main diagonal: A000894.
Sums: (-1)^floor((n+1)/2)*A001405 (signed row), A001700 (row), A203611 (diagonal).
Cf. A001263, A007318, A127648, A281260.
Cf. A103371 (mirrored).

Flat(List([0..10],n->List([0..n], k->(k+1)*Binomial(n+1,k+1)*Binomial(n+1,k)/(n+1)))); # Muniru A Asiru, Feb 26 2019

a132813 n k = a132813_tabl !! n !! k
a132813_row n = a132813_tabl !! n
a132813_tabl = zipWith (zipWith (*)) a007318_tabl $ tail a007318_tabl
-- Reinhard Zumkeller, Apr 04 2014

/* triangle */ [[(k+1)*Binomial(n+1,k+1)*Binomial(n+1,k)/(n+1): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Oct 19 2014

P := (n, x) -> hypergeom([1-n, -n], [1], x): for n from 1 to 9 do PolynomialTools:-CoefficientList(simplify(P(n,x)),x) od; # Peter Luschny, Nov 26 2014

T[n_,k_]=Binomial[n-1,k-1]*Binomial[n,k-1]; Table[Table[T[n,k],{k,1,n}],{n,1,11}]; Flatten[%] (* Roger L. Bagula, Apr 09 2008 *)
P[n_, x_] := HypergeometricPFQ[{1-n, -n}, {1}, x]; Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten (* Jean-François Alcover, Nov 27 2014, after Peter Luschny *)

tabl(nn) = {for (n = 1, nn, for (k = 1, n, print1(binomial(n-1, k-1)*binomial(n, k-1) , ", ");););} \\ Michel Marcus, Feb 12 2014

def A132813(n,k): return binomial(n,k)*binomial(n+1,k)
print(flatten([[A132813(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Mar 12 2025

T(n,k) = (k+1)*binomial(n+1,k+1)*binomial(n+1,k)/(n+1), n >= k >= 0.
From Roger L. Bagula, May 14 2010: (Start)
T(n, m) = coefficients(p(x,n)), where
p(x,n) = (1-x)^(2*n)*Sum_{k >= 0} binomial(k+n-1, k)*binomial(n+k, k)*x^k,
 or p(x,n) = (1-x)^(2*n)*Hypergeometric2F1([n, n+1], [1], x). (End)
T(n,k) = binomial(n,k) * binomial(n+1,k). - Reinhard Zumkeller, Apr 04 2014
These are the coefficients of the polynomials Hypergeometric2F1([1-n,-n], [1], x). - Peter Luschny, Nov 26 2014
G.f.: A(x,y) = A281260(x,y)/(1-A281260(x,y))/x. - Vladimir Kruchinin, Oct 10 2020

A281293 Triangular array of generalized Narayana Numbers T(n,k) = 3binomial(n+1,k) binomial(n-3,k-1)/(n+1) for n >= 2 and 0 <= k <= n-2, read by rows.

Original entry on oeis.org

1, 0, 3, 0, 3, 6, 0, 3, 15, 10, 0, 3, 27, 45, 15, 0, 3, 42, 126, 105, 21, 0, 3, 60, 280, 420, 210, 28, 0, 3, 81, 540, 1260, 1134, 378, 36, 0, 3, 105, 945, 3150, 4410, 2646, 630, 45, 0, 3, 132, 1540, 6930, 13860, 12936, 5544, 990, 55, 0, 3, 162, 2376, 13860, 37422, 49896, 33264, 10692, 1485, 66

Offset: 2

Werner Schulte, Jan 19 2017

The current array is the case m = 2 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 = 3 see A281297.

			The triangle begins:
n\k:  0  1    2     3      4      5       6       7      8      9    10  11  ...
02 :  1
03 :  0  3
04 :  0  3    6
05 :  0  3   15    10
06 :  0  3   27    45     15
07 :  0  3   42   126    105     21
08 :  0  3   60   280    420    210      28
09 :  0  3   81   540   1260   1134     378      36
10 :  0  3  105   945   3150   4410    2646     630     45
11 :  0  3  132  1540   6930  13860   12936    5544    990     55
12 :  0  3  162  2376  13860  37422   49896   33264  10692   1485    66
13 :  0  3  195  3510  25740  90090  162162  154440  77220  19305  2145  78
etc.

David Callan, Generalized Narayana Numbers

Cf. A000007, A001263, A033184, A281260, A281297.

Table[3 Binomial[n + 1, k] Binomial[n - 3, k - 1]/(n + 1), {n, 2, 12}, {k, 0, n - 2}] // Flatten (* Michael De Vlieger, Jan 19 2017 *)

Row sums are A033184(n+1,3).

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

A132813 Triangle read by rows: A001263 * A127648 as infinite lower triangular matrices.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Haskell

Magma

Maple

Mathematica

PARI

SageMath

Formula

A281293 Triangular array of generalized Narayana Numbers T(n,k) = 3binomial(n+1,k) binomial(n-3,k-1)/(n+1) for n >= 2 and 0 <= k <= n-2, read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

cp's OEIS Frontend

A132813 Triangle read by rows: A001263 * A127648 as infinite lower triangular matrices.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Haskell

Magma

Maple

Mathematica

PARI

SageMath

Formula

A281293 Triangular array of generalized Narayana Numbers T(n,k) = 3*binomial(n+1,k)* binomial(n-3,k-1)/(n+1) for n >= 2 and 0 <= k <= n-2, 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

A281293 Triangular array of generalized Narayana Numbers T(n,k) = 3binomial(n+1,k) binomial(n-3,k-1)/(n+1) for n >= 2 and 0 <= k <= n-2, 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.