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A132813 Triangle read by rows: A001263 * A127648 as infinite lower triangular matrices. 7
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, 1, 56, 588, 1960, 2450, 1176, 196, 8, 1, 72, 1008, 4704, 8820, 7056, 2352, 288, 9, 1, 90, 1620, 10080, 26460, 31752, 17640, 4320, 405, 10 (list; table; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

CROSSREFS

Cf. A127648, A001263, A001700.

Cf. A007318, A000894 (central terms), A103371 (mirrored).

Sequence in context: A120108 A060556 A222969 * A034898 A059300 A321331

Adjacent sequences:  A132810 A132811 A132812 * A132814 A132815 A132816

KEYWORD

nonn,tabl

AUTHOR

Gary W. Adamson, Sep 01 2007

STATUS

approved

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Last modified February 16 12:48 EST 2019. Contains 320163 sequences. (Running on oeis4.)