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

1,2

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

Links

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

Peter Luschny, Meanders.

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

Example

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

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 *)

Crossrefs

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

Sequence in context: A184844 A252848 A241475 * A203371 A181206 A274959

Adjacent sequences: A132809 A132810 A132811 * A132813 A132814 A132815

Keyword

nonn,tabl

Author

Gary W. Adamson, Sep 01 2007

Extensions

New name from Peter Luschny, Dec 19 2011

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

Status

approved