A103406 Triangle read by rows: n-th row = unsigned coefficients of the characteristic polynomials of an n X n matrix with 2's on the diagonal and 1's elsewhere.

1, 1, 2, 1, 4, 3, 1, 6, 9, 4, 1, 8, 18, 16, 5, 1, 10, 30, 40, 25, 6, 1, 12, 45, 80, 75, 36, 7, 1, 14, 63, 140, 175, 126, 49, 8, 1, 16, 84, 224, 350, 336, 196, 64, 9, 1, 18, 108, 336, 630, 756, 588, 288, 81, 10, 1, 20, 135, 480, 1050, 1512, 1470, 960, 405, 100, 11, 1, 22, 165

Offset: 0

Author

Gary W. Adamson, Feb 04 2005

Keywords

• nonn
• tabl

Comments

This triangle * [1/1, 1/2, 1/3, ...] = (1, 2, 4, 8, 16, 32, ...). - Gary W. Adamson, Nov 15 2007

Triangle read by rows: T(n,k) = (k+1)*binomial(n,k), 0 <= k <= n. - Philippe Deléham, Apr 20 2009

Examples

			Characteristic polynomial of 3 X 3 matrix [2 1 1 / 1 2 1 / 1 1 2] = x^3 - 6x^2 + 9x - 4.
The first few characteristic polynomials are:
  1
  x - 2
  x^2 - 4x + 3
  x^3 - 6x^2 + 9x - 4
  x^4 - 8x^3 + 18x^2 - 16x + 5
		

Links

• Vincenzo Librandi, Rows n = 0..100, flattened

Crossrefs

Row sums = A001792: 1, 3, 8, 20, 48, 112, ...

See A103283 for the mirror image.

Cf. A093375, A127648, A128064.

Programs

Maple

with(linalg): printf(`%d,`,1): for n from 1 to 15 do mymat:=array(1..n, 1..n): for i from 1 to n do for j from 1 to n do if i=j then mymat[i,j]:=2 else mymat[i,j]:=1 fi: od: od: temp:=charpoly(mymat,x): for j from n to 0 by -1 do printf(`%d,`,abs(coeff(temp, x, j))) od: od: # James Sellers, Apr 22 2005
p := (n,x) -> (x+1)^(n-1)+(x+1)^(n-2)*(n-1);
seq(seq(coeff(p(n,x),x,n-j-1),j=0..n-1),n=1..11); # Peter Luschny, Feb 25 2014

Mathematica

t[n_, k_] := (k+1)*Binomial[n, k]; Table[t[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 09 2012, after Philippe Deléham *)

Formula

Binomial transform of A127648. - Gary W. Adamson, Nov 15 2007

Equals A128064 * A007318. - Gary W. Adamson, Jan 03 2008

T(n,k) = (k+1)*A007318(n,k). - Philippe Deléham, Apr 20 2009

T(n,k) = Sum_{i=1..k+1} i*binomial(k+1,i)*binomial(n-k,k+1-i). - Mircea Merca, Apr 11 2012

O.g.f.: (1 - y)/(1 - y - x*y)^2 = 1 + (1 + 2*x)*y + (1 + 4*x + 3*x*2)*y^2 + .... - Peter Bala, Oct 18 2023

Extensions

More terms from James Sellers, Apr 22 2005