A132813 - OEIS

A132813

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

%I #69 Feb 09 2026 04:45:56

%S 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,

%T 525,126,7,1,56,588,1960,2450,1176,196,8,1,72,1008,4704,8820,7056,

%U 2352,288,9,1,90,1620,10080,26460,31752,17640,4320,405,10

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

%C 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

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

%H Reinhard Zumkeller, <a href="/A132813/b132813.txt">Rows n = 0..125 of table, flattened</a>

%H N. Alexeev and A. Tikhomirov, <a href="https://arxiv.org/abs/1501.04615">Singular Values Distribution of Squares of Elliptic Random Matrices and type-B Narayana Polynomials</a>, arXiv preprint arXiv:1501.04615 [math.PR], 2015.

%H C. Athanasiadis and C. Savvidou, <a href="https://arxiv.org/abs/1204.0362">The local h-vector of the cluster subdivision of a simplex</a>, arXiv preprint arXiv:1204.0362 [math.CO], 2012.

%H Robert. A. Sulanke, <a href="https://doi.org/10.37236/1518">Counting Lattice Paths by Narayana Polynomials</a> Electronic J. Combinatorics 7, No. 1, R40, 1-9, 2000.

%H Hua Xin and Huan Xiong, <a href="https://ajc.maths.uq.edu.au/pdf/94/ajc_v94_p177.pdf">Descents in the Grand Dyck paths and the Chung-Feller property</a>, Australas. J. Combin. 94 (1) (2026), 177-194. See Table 1 at page 192.

%F T(n,k) = (k+1)*binomial(n+1,k+1)*binomial(n+1,k)/(n+1), n >= k >= 0.

%F From _Roger L. Bagula_, May 14 2010: (Start)

%F T(n, m) = coefficients(p(x,n)), where

%F p(x,n) = (1-x)^(2*n)*Sum_{k >= 0} binomial(k+n-1, k)*binomial(n+k, k)*x^k,

%F or p(x,n) = (1-x)^(2*n)*Hypergeometric2F1([n, n+1], [1], x). (End)

%F T(n,k) = binomial(n,k) * binomial(n+1,k). - _Reinhard Zumkeller_, Apr 04 2014

%F These are the coefficients of the polynomials Hypergeometric2F1([1-n,-n], [1], x). - _Peter Luschny_, Nov 26 2014

%F G.f.: A(x,y) = A281260(x,y)/(1-A281260(x,y))/x. - _Vladimir Kruchinin_, Oct 10 2020

%e First few rows of the triangle are:

%e 1;

%e 1, 2;

%e 1, 6, 3;

%e 1, 12, 18, 4;

%e 1, 20, 60, 40, 5;

%e 1, 30, 150, 200, 75, 6;

%e 1, 42, 315, 700, 525, 126, 7;

%e ...

%p 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 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 *)

%t 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_ *)

%o (PARI) 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

%o (Haskell)

%o a132813 n k = a132813_tabl !! n !! k

%o a132813_row n = a132813_tabl !! n

%o a132813_tabl = zipWith (zipWith (*)) a007318_tabl $ tail a007318_tabl

%o -- _Reinhard Zumkeller_, Apr 04 2014

%o (Magma) /* 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

%o (GAP) 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

%o (SageMath)

%o def A132813(n,k): return binomial(n,k)*binomial(n+1,k)

%o print(flatten([[A132813(n,k) for k in range(n+1)] for n in range(13)])) # _G. C. Greubel_, Mar 12 2025

%Y Family of polynomials (see A062145): A008459 (c=1), this sequence (c=2), A062196 (c=3), A062145 (c=4), A062264 (c=5), A062190 (c=6).

%Y 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).

%Y Diagonals: A000027 (k=n), A002411 (k=n-1), A004302 (k=n-2), A108647 (k=n-3), A134287 (k=n-4).

%Y Main diagonal: A000894.

%Y Sums: (-1)^floor((n+1)/2)*A001405 (signed row), A001700 (row), A203611 (diagonal).

%Y Cf. A001263, A007318, A127648, A281260.

%Y Cf. A103371 (mirrored).

%K nonn,tabl

%O 0,3

%A _Gary W. Adamson_, Sep 01 2007