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

%I A132813 #64 Apr 08 2025 08:47:34
%S A132813 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 A132813 525,126,7,1,56,588,1960,2450,1176,196,8,1,72,1008,4704,8820,7056,
%U A132813 2352,288,9,1,90,1620,10080,26460,31752,17640,4320,405,10
%N A132813 Triangle read by rows: A001263 * A127648 as infinite lower triangular matrices.
%C A132813 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 A132813 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 A132813 Reinhard Zumkeller, <a href="/A132813/b132813.txt">Rows n = 0..125 of table, flattened</a>
%H A132813 N. Alexeev and 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 A132813 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 A132813 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.
%F A132813 T(n,k) = (k+1)*binomial(n+1,k+1)*binomial(n+1,k)/(n+1), n >= k >= 0.
%F A132813 From _Roger L. Bagula_, May 14 2010: (Start)
%F A132813 T(n, m) = coefficients(p(x,n)), where
%F A132813 p(x,n) = (1-x)^(2*n)*Sum_{k >= 0} binomial(k+n-1, k)*binomial(n+k, k)*x^k,
%F A132813  or p(x,n) = (1-x)^(2*n)*Hypergeometric2F1([n, n+1], [1], x). (End)
%F A132813 T(n,k) = binomial(n,k) * binomial(n+1,k). - _Reinhard Zumkeller_, Apr 04 2014
%F A132813 These are the coefficients of the polynomials Hypergeometric2F1([1-n,-n], [1], x). - _Peter Luschny_, Nov 26 2014
%F A132813 G.f.: A(x,y) = A281260(x,y)/(1-A281260(x,y))/x. - _Vladimir Kruchinin_, Oct 10 2020
%e A132813 First few rows of the triangle are:
%e A132813   1;
%e A132813   1,  2;
%e A132813   1,  6,   3;
%e A132813   1, 12,  18,   4;
%e A132813   1, 20,  60,  40,   5;
%e A132813   1, 30, 150, 200,  75,   6;
%e A132813   1, 42, 315, 700, 525, 126, 7;
%e A132813   ...
%p A132813 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 A132813 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 A132813 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 A132813 (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 A132813 (Haskell)
%o A132813 a132813 n k = a132813_tabl !! n !! k
%o A132813 a132813_row n = a132813_tabl !! n
%o A132813 a132813_tabl = zipWith (zipWith (*)) a007318_tabl $ tail a007318_tabl
%o A132813 -- _Reinhard Zumkeller_, Apr 04 2014
%o A132813 (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 A132813 (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 A132813 (SageMath)
%o A132813 def A132813(n,k): return binomial(n,k)*binomial(n+1,k)
%o A132813 print(flatten([[A132813(n,k) for k in range(n+1)] for n in range(13)])) # _G. C. Greubel_, Mar 12 2025
%Y A132813 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 A132813 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 A132813 Diagonals: A000027 (k=n), A002411 (k=n-1), A004302 (k=n-2), A108647 (k=n-3), A134287 (k=n-4).
%Y A132813 Main diagonal: A000894.
%Y A132813 Sums: (-1)^floor((n+1)/2)*A001405 (signed row), A001700 (row), A203611 (diagonal).
%Y A132813 Cf. A001263, A007318, A127648, A281260.
%Y A132813 Cf. A103371 (mirrored).
%K A132813 nonn,tabl
%O A132813 0,3
%A A132813 _Gary W. Adamson_, Sep 01 2007