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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. 8
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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

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

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

EXAMPLE

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

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

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

See A103283 for the mirror image.

Cf. A093375, A127648, A128064.

Sequence in context: A210559 A180803 A093966 * A142978 A152060 A093190

Adjacent sequences:  A103403 A103404 A103405 * A103407 A103408 A103409

KEYWORD

nonn,tabl

AUTHOR

Gary W. Adamson, Feb 04 2005

EXTENSIONS

More terms from James A. Sellers, Apr 22 2005

STATUS

approved

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Last modified August 5 14:29 EDT 2021. Contains 346469 sequences. (Running on oeis4.)