A132812 Triangle read by rows, n>=1, 1<=k<=n, T(n,k) = k*binomial(n,k)^2/(n-k+1).

1, 2, 2, 3, 9, 3, 4, 24, 24, 4, 5, 50, 100, 50, 5, 6, 90, 300, 300, 90, 6, 7, 147, 735, 1225, 735, 147, 7, 8, 224, 1568, 3920, 3920, 1568, 224, 8, 9, 324, 3024, 10584, 15876, 10584, 3024, 324, 9, 10, 450, 5400, 25200, 52920, 52920, 25200, 5400, 450, 10

Offset: 1

Author

Gary W. Adamson, Sep 01 2007

Keywords

• nonn
• tabl

Comments

A127648 * A001263. (Original name by Gary W. Adamson.)

Let a meander be defined as in the link and m = 2. Then T(n,k) counts the invertible meanders of length m(n+1) built from arcs with central angle 360/m whose binary representation have mk '1's. - Peter Luschny, Dec 19 2011

Antidiagonal sums = A110320. - Philippe Deléham, Jun 08 2013

Examples

			First few rows of the triangle are:
  1;
  2, 2;
  3, 9, 3;
  4, 24, 24, 4;
  5, 50, 100, 50, 5;
  6, 90, 300, 300, 90, 6;
  ...
Row 4 = (4, 24, 24, 4) = 4 * (1, 6, 6, 1), where (1, 6, 6, 1) = row 4 of the Narayana triangle. - _Gary W. Adamson_
T(3,1) = 3 because the invertible meanders of length 8 and central angle 180 degree which have two '1's in their binary representation are {10000100, 10010000, 11000000}. - _Peter Luschny_, Dec 19 2011
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows 1..150).
• Peter Luschny, Meanders.

Crossrefs

Row sums: A001791. Cf. A001263, A127648, A001791, A202409.

Programs

Maple

A132812 := (n,k) -> k*binomial(n,k)^2/(n-k+1);
seq(print(seq(A132812(n,k),k=0..n-1)),n=1..6); # Peter Luschny, Dec 19 2011

Mathematica

Table[k Binomial[n, k]^2/(n - k + 1), {n, 10}, {k, n}] // Flatten (* Michael De Vlieger, Nov 15 2017 *)

Formula

A127648 * A001263 as infinite lower triangular matrices.

a(n) = n * A001263(n,k).

T(n,k) = binomial(n,k)*binomial(n,k-1). - Philippe Deléham, Jun 08 2013

G.f.: x*d(N(x,y))/dx, where N(x,y) is g.f. for Narayana numbers A001263. - Vladimir Kruchinin, Oct 22 2021

Extensions

New name from Peter Luschny, Dec 19 2011

a(53) corrected by Michael De Vlieger, Nov 15 2017