OFFSET
0,3
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
LINKS
Vincenzo Librandi, Rows n = 0..100, flattened
FORMULA
Binomial transform of A127648. - Gary W. Adamson, Nov 15 2007
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
EXAMPLE
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
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 A. 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 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, Feb 04 2005
EXTENSIONS
More terms from James A. Sellers, Apr 22 2005
STATUS
approved