OFFSET
0,3
COMMENTS
Row sums = A001700: (1, 3, 10, 35, 126, ...).
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
h-vector for cluster complex associated to the root system B_n. See p. 8, Athanasiadis and C. Savvidou. - Tom Copeland, Oct 19 2014
LINKS
Reinhard Zumkeller, Rows n = 0..125 of table, flattened
N. Alexeev, 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.
FORMULA
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)
p(x,n) = (1 - x)^(2*n)*Sum[Binomial[k + n - 1, k]*Binomial[n + k, k]*x^k, {k, 0, Infinity}];
p(x,n) = (1 - x)^(2n) HypergeometricPFQ[{n, 1 + n}, {1}, x];
t(n,m) = coefficients(p(x,n)) (End)
T(n,k) = binomial(n,k) * binomial(n+1,k). - Reinhard Zumkeller, Apr 04 2014
These are the coefficients of the polynomials hypergeom([1-n,-n],[1],x). - Peter Luschny, Nov 26 2014
EXAMPLE
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,
...
MAPLE
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
MATHEMATICA
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 *)
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 *)
PROG
(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
(Haskell)
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
(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
(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
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, Sep 01 2007
STATUS
approved