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%I #55 Feb 13 2024 07:53:16
%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 Row sums = A001700: (1, 3, 10, 35, 126, ...).
%C Also a(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, A. Tikhomirov, <a href="http://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="http://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="http://www.combinatorics.org/Volume_7/Abstracts/v7i1r40.html">Counting Lattice Paths by Narayana Polynomials</a> Electronic J. Combinatorics 7, No. 1, R40, 1-9, 2000.
%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 p(x,n) = (1 - x)^(2*n)*Sum[Binomial[k + n - 1, k]*Binomial[n + k, k]*x^k, {k, 0, Infinity}];
%F p(x,n) = (1 - x)^(2n) HypergeometricPFQ[{n, 1 + n}, {1}, x];
%F t(n,m) = coefficients(p(x,n)) (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 hypergeom([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 A[n_,k_]=Binomial[n-1,k-1]*Binomial[n,k-1]; Table[Table[A[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
%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 Cf. A127648, A001263, A001700.
%Y Cf. A007318, A000894 (central terms), A103371 (mirrored).
%K nonn,tabl
%O 0,3
%A _Gary W. Adamson_, Sep 01 2007
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