A103406 - OEIS

%I #39 Oct 19 2023 12:52:15

%S 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,

%T 14,63,140,175,126,49,8,1,16,84,224,350,336,196,64,9,1,18,108,336,630,

%U 756,588,288,81,10,1,20,135,480,1050,1512,1470,960,405,100,11,1,22,165

%N 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.

%C This triangle * [1/1, 1/2, 1/3, ...] = (1, 2, 4, 8, 16, 32, ...). - _Gary W. Adamson_, Nov 15 2007

%C Triangle read by rows: T(n,k) = (k+1)*binomial(n,k), 0 <= k <= n. - _Philippe Deléham_, Apr 20 2009

%H Vincenzo Librandi, <a href="/A103406/b103406.txt">Rows n = 0..100, flattened</a>

%F Binomial transform of A127648. - _Gary W. Adamson_, Nov 15 2007

%F Equals A128064 * A007318. - _Gary W. Adamson_, Jan 03 2008

%F T(n,k) = (k+1)*A007318(n,k). - _Philippe Deléham_, Apr 20 2009

%F T(n,k) = Sum_{i=1..k+1} i*binomial(k+1,i)*binomial(n-k,k+1-i). - _Mircea Merca_, Apr 11 2012

%F 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

%e Characteristic polynomial of 3 X 3 matrix [2 1 1 / 1 2 1 / 1 1 2] = x^3 - 6x^2 + 9x - 4.

%e The first few characteristic polynomials are:

%e   1

%e   x - 2

%e   x^2 - 4x + 3

%e   x^3 - 6x^2 + 9x - 4

%e   x^4 - 8x^3 + 18x^2 - 16x + 5

%p 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 p := (n,x) -> (x+1)^(n-1)+(x+1)^(n-2)*(n-1);

%p seq(seq(coeff(p(n,x),x,n-j-1),j=0..n-1),n=1..11); # _Peter Luschny_, Feb 25 2014

%t 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_ *)

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

%Y See A103283 for the mirror image.

%Y Cf. A093375, A127648, A128064.

%K nonn,tabl

%O 0,3

%A _Gary W. Adamson_, Feb 04 2005

%E More terms from _James A. Sellers_, Apr 22 2005