OFFSET
0,3
COMMENTS
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 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.
Hua Xin and Huan Xiong, Descents in the Grand Dyck paths and the Chung-Feller property, Australas. J. Combin. 94 (1) (2026), 177-194. See Table 1 at page 192.
FORMULA
T(n,k) = (k+1)*binomial(n+1,k+1)*binomial(n+1,k)/(n+1).
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)
These are the coefficients of the polynomials Hypergeometric2F1([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
T[n_, k_]=Binomial[n, k]*Binomial[n+1, k]; Table[T[n, k], {n, 0, 11}, {k, 0, n}]//Flatten (* Roger L. Bagula, Apr 09 2008; corrected by Stefano Spezia, Jul 31 2026 *)
(* Alternative: *)
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 *)
(* Alternative: *)
T[n_, 0] /; n>=0 := 1; T[n_, k_] /; 0<k<=n := T[n, k] = n (n+1)/(k k) T[n-1, k-1];
T[n_, k_] := 0; Flatten@Table[T[n, k], {n, 0, 9}, {k, 0, n}] (* Oliver Seipel, Jul 31 2026 *)
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
(SageMath)
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
CROSSREFS
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).
Main diagonal: A000894.
Cf. A103371 (mirrored).
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, Sep 01 2007
STATUS
approved