A136678 -id:A136678 - cp's OEIS frontend

A143717 (1, 2, 3, 2^2, 5, 23, 7, 2^3, 3^2, 25, 11, 2^23, 13, 27, 3*5,..) becomes (abs(1-2-3, 2-2-5, 2-3-7, 2-3-3, 2-2-5, 11-2-2, 3-13-2, 7-3-5,..)).

4, 5, 8, 4, 5, 7, 12, 1, 19, 3, 15, 5, 32, 4, 1, 7, 7, 24, 28, 9, 20, 3, 36, 20, 8, 38, 47, 11, 4, 68, 5, 3, 0, 13, 42, 5, 6, 15, 86, 3, 58, 26, 1, 10, 75, 17, 22, 73, 4, 73, 29, 2, 1, 14, 73, 2, 122, 3, 15, 44, 24, 80, 4, 8, 24, 18, 16, 95, 7, 10, 5, 101, 117, 14, 9, 158, 3, 108, 9, 31

Offset: 1

Juri-Stepan Gerasimov, Nov 13 2008

			Abs (7-3-5)=abs (-1) =1=a(8).
Abs (2-4-17)=abs(-19)=19=a(9).
Abs (2-3-2)=abs (-3)=3=a(10).
Abs (19-2-2)=abs 15=15=a(11),
Abs (5-3-7)=abs (-5)=5=a(12),
etc.

Cf. A141287, A141261, A136678.

Replaced a 56 by a 1. R. J. Mathar, Feb 21 2009

A143852 (1, 2, 3, 2^2, 5, 23, 7, 2^3, 3^2, 25, 11, 2^23, 13, 27, 3*5,..) becomes (abs(1-2-3-2, 2-5-2-3, 7-2-3-3, 2-2-5-11, 2-2-3-13, 2-7-3-5,..)).

Original entry on oeis.org

6, 8, 1, 16, 16, 13, 21, 20, 13, 34, 7, 17, 36, 37, 17, 27, 5, 13, 3, 29, 28, 6, 30, 8, 17, 66, 9, 1, 47, 52, 89, 8, 14, 77, 41, 81, 6, 31, 1, 1, 91, 8, 124, 13, 31, 13, 82, 17, 55, 69, 103, 10, 13, 100, 119, 18, 155, 6, 91, 40, 111, 11, 32, 55, 9, 10, 15, 37, 10, 115, 23, 124, 17

Offset: 1

Juri-Stepan Gerasimov, Nov 17 2008

nonn

			Abs (2-7-3-5)=abs (-13) =13=a(6).
Abs (2-4-17-2)=abs(-21)=21=a(7).
Abs (3-2-19-2)=abs (-20)=20=a(8).
Abs (2-5-3-7)=abs (-13)=13=a(9).
Abs (2-11-23-2)=abs(-34)=34=a(10),
etc.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A141287, A141261, A136678, A143717.

g:= proc(n) local F;
   F:= sort(ifactors(n)[2], (a,b) -> a[1]Robert Israel, Apr 25 2016

Abs@ Total@ MapAt[Abs, Minus@ #, {1}] & /@ Partition[#, 4] &@ Flatten[FactorInteger /@ Range@ 134 /. {a_, b_} /; b == 1 :> {a}] (* Michael De Vlieger, Apr 25 2016 *)

Corrected (10 replaced by 8) and extended by R. J. Mathar, Apr 18 2010

A123948 Triangle read by rows: first row is 1, and n-th row (n > 0) gives the coefficients in the expansion of the characteristic polynomial of the (n - 1)-th Bernstein basis matrix, horizontal flipped.

Original entry on oeis.org

1, 1, -1, -1, 1, 1, -2, 3, 3, -1, 9, -15, -22, 7, 1, 96, -184, -314, 139, 19, -1, -2500, 5250, 10575, -5375, -1026, 51, 1, -162000, 369900, 842310, -498171, -111179, 7644, 141, -1, 26471025, -64790985, -164634169, 109325076, 28870212, -2322404, -59193, 393, 1

Offset: 0

Roger L. Bagula and Gary W. Adamson, Oct 26 2006

The Bernstein basis matrix of order n - 1 is an n X n matrix whose m-th row represents the coefficients in the expansion of the Bernstein basis polynomial defined as binomial(n, m)*x^m*(1 - x)^(n - m), 0 <= m <= n - 1. For n = 0, we obtain the 0 X 0 matrix. The convention is that the characteristic polynomial of the empty matrix is identically 1 (see [de Boor] and [Johnson et al.]). Row n of the present sequence is obtained by taking the characteristic polynomial of the matrix represented by the polynomials binomial(n, m)*x^(n - m)*(1 - x)^m. The resulting matrix is, in fact, the horizontal flipped version of the Bernstein basis matrix of order n (see example). - Franck Maminirina Ramaharo, Oct 19 2018

			Triangle begins:
        1;
        1,     -1;
       -1,      1,      1;
       -2,      3,      3,      -1;
        9,    -15,    -22,       7,       1;
       96,   -184,   -314,     139,      19,   -1;
    -2500,   5250,  10575,   -5375,   -1026,   51,   1;
  -162000, 369900, 842310, -498171, -111179, 7644, 141, -1;
      ...
From _Franck Maminirina Ramaharo_, Oct 19 2018: (Start)
Let n = 6 (i.e., order 5). The corresponding Bernstein basis matrix has the form
   1, -5,  10, -10,   5,  -1
   0,  5, -20,  30, -20,   5
   0,  0,  10, -30,  30, -10
   0,  0,   0,  10, -20,  10
   0,  0,   0,   0,   5,  -5
   0,  0,   0,   0,   0,   1.
Flipping this matrix horizontally gives the matrix for the polynomials binomial(5, m)*x^(5 - m)*(1 - x)^m, 0 <= m <= 5,
   0,  0,   0,   0,   0,   1
   0,  0,   0,   0,   5,  -5
   0,  0,   0,  10, -20,  10
   0,  0,  10, -30,  30, -10
   0,  5, -20,  30, -20,   5
   1, -5,  10, -10,   5,  -1
whose characteristic polynomial is -2500 + 5250*x + 10575*x^2 - 5375*x^3 - 1026*x^4 + 51*x^5 + x^6. (End)

Gengzhe Chang and Thomas W. Sederberg, Over and Over Again, The Mathematical Association of America, 1997, Chap. 30.

Carl de Boor, An empty exercise
John Burkardt, BERNSTEIN_POLYNOMIAL - The Bernstein Polynomials
Charles R. Johnson and Carlos M. Saiago, Eigenvalues, Multiplicities and Graphs, Cambridge University Press, 2018, p. 8.
Kenneth I. Joy, On-Line Geometric Modeling Notes
Wikipedia, Bernstein polynomial

Cf. A045912, A136678.

M[n_] := Table[CoefficientList[Binomial[n - 1, n - i - 1]*(1 - x)^i*x^(n - i - 1), x], {i, 0, n - 1}];
Join[{1}, Table[CoefficientList[CharacteristicPolynomial[M[d], x], x], {d, 1, 10}]]//Flatten

Edited, new name, offset corrected by Franck Maminirina Ramaharo, Oct 19 2018

A144022 (1, 2, 3, 2^2, 5, 23, 7, 2^3, 3^2, 25, 11, 2^23, 13, 27, 3*5,..) becomes (abs(1-2+3-2, 2-5+2-3, 7-2+3-3, 2-2+5-11, 2-2+3-13, 2-7+3-5,..)).

Original entry on oeis.org

0, 4, 5, 6, 10, 7, 13, 18, 7, 12, 3, 11, 22, 27, 11, 17, 1, 51, 9, 35, 32, 4, 26, 4, 11, 40, 3, 5, 43, 56, 85, 6, 12, 57, 35, 67, 0, 105, 3, 21, 67, 2, 42, 1, 55, 19, 88, 9, 49, 59, 97, 6, 9, 102, 87, 12, 49, 0, 101, 36, 107, 35, 26, 63, 15, 6, 97, 27, 6, 129, 33, 128, 21, 62, 121

Offset: 1

Juri-Stepan Gerasimov, Nov 17 2008

nonn

			Abs (2-7+3-5)=abs (-7) =7=a(6).
Abs (2-4+17-2)=abs(13)=13=a(7).
Abs (3-2+19-2)=abs (18)=18=a(8).
Abs (2-5+3-7)=abs (-7)=7=a(9).
Abs (2-11+23-2)=abs(12)=12=a(10),
etc.

Cf. A141287, A141261, A136678, A143717.

Corrected (11 replaced by 9, 5 replaced by 6, 104 by 102) by R. J. Mathar, Apr 18 2010

A144076 (1, 2, 3, 2^2, 5, 23, 7, 2^3, 3^2, 25, 11, 2^23, 13, 27, 35,..) becomes (abs(1-23-2, 2-52-3, 7-23-3, 2-25-11, 2-23-13, 2-7*3-5,..)).

Original entry on oeis.org

7, 11, 2, 19, 17, 24, 68, 37, 20, 253, 14, 40, 41, 44, 24, 90, 6, 4, 2, 28, 28, 9, 75, 13, 18, 77, 12, 2, 64, 52, 148, 13, 61, 736, 72, 104, 9, 4, 1, 60, 1026, 15, 3403, 24, 72, 12, 81, 88, 98, 252, 110, 15, 22, 199, 1750, 41, 206, 7, 88, 75, 222, 98, 87, 112, 8, 19, 64, 156

Offset: 1

Juri-Stepan Gerasimov, Nov 17 2008

nonn

			Abs (2-7*3-5)=abs (-24) =24=a(6).
Abs (2-4*17-2)=abs(-68)=68=a(7).
Abs (3-2*19-2)=abs (-37)=37=a(8).
Abs (2-5*3-7)=abs (-20)=20=a(9).
Abs (2-11*23-2)=abs(-253)=253=a(10),
etc.

Cf. A141287, A141261, A136678, A143717.

Corrected (12 replaced by 52) by R. J. Mathar, Apr 18 2010

A144114 (1, 2, 3, 2^2, 5, 23, 7, 2^3, 3^2, 25, 11, 2^23, 13, 27, 3*5,..) becomes (abs(1-2^3-2, 2-5^2-3, 7-2^3-3, 2-2^5-11, 2-2^3-13, 2-7^3-5,..)).

Original entry on oeis.org

9, 26, 4, 41, 19, 346, 17179869184, 524287, 130, 895430243255237372246531, 242, 2198, 155, 272, 134, 1419862, 8, 524254, 0, 26, 28, 31, 2190, 48, 20, 8243, 30, 4, 387, 52, 3747, 127, 1220703121

Offset: 1

Juri-Stepan Gerasimov, Nov 17 2008

nonn

			Abs (2-7^3-5)=abs (-346) =346=a(6).
Abs (2-4^17-2)=abs(-17179869184)=17179869184=a(7).
Abs (3-2^19-2)=abs (-524287)=524287=a(8).
Abs (2-5^3-7)=abs (-130)=130=a(9),
etc.

Cf. A141287, A141261, A136678, A143717.

More terms from R. J. Mathar, Apr 18 2010

A173820 Coefficients of characteristic polynomials of Hadamard Cartan F_2 self-similar 2^n matrices:M={{2, -1}, {-2, 2}}.

Original entry on oeis.org

1, 2, -4, 1, 16, -64, 56, -16, 1, 4096, -32768, 75776, -77824, 39296, -9728, 1184, -64, 1, 4294967296, -68719476736, 375809638400, -1043677052928, 1696981843968, -1726845288448, 1143073669120, -506453819392, 152912134144, -31653363712

Offset: 0

Roger L. Bagula, Feb 25 2010

Row sums are:
{1, -1, -7, -31, -208289151, 199276356275696712709633,
-27294457550222463310332530871924308277403810665846783,...}.

			{1},
{2, -4, 1},
{ 16, -64, 56, -16, 1},
{4096, -32768, 75776, -77824, 39296, -9728, 1184, -64, 1}, ...

Cf. A136674, A158800, A173814, A136678

Clear[HadamardMatrix];
MatrixJoinH[A_, B_] := Transpose[Join[Transpose[A], Transpose[B]]]
KroneckerProduct[M_, N_] := Module[{M1, N1, LM, LN, N2},
M1 = M;
N1 = N;
LM = Length[M1];
LN = Length[N1];
Do[M1[[i, j]] = M1[[i, j]]N1, {i, 1, LM}, {j, 1, LM}];
Do[M1[[i, 1]] = MatrixJoinH[M1[[i, 1]], M1[[i, j]]], {j, 2, LM}, {i, 1, LM}];
N2 = {};
Do[AppendTo[N2, M1[[i, 1]]], {i, 1, LM}];
N2 = Flatten[N2];
Partition[N2, LM*LN, LM*LN]]
HadamardMatrix[2] := {{2, -1}, {-2, 2}}
HadamardMatrix[n_] := Module[{m},
m = {{2, -1}, {-2, 2}};
KroneckerProduct[m, HadamardMatrix[n/2]]]
Table[HadamardMatrix[2^n], {n, 1, 4}]
Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[ HadamardMatrix[2^n], x], x], {n, 1, 6}]]
Flatten[%]

M(2)={{2, -1}, {-2, 2}};

M(4)={{4, -2, -2, 1}, {-4, 4, 2, -2}, {-4, 2, 4, -2}, {4, -4, -4, 4}},etc.

cp's OEIS Frontend

A143717 (1, 2, 3, 2^2, 5, 2*3, 7, 2^3, 3^2, 2*5, 11, 2^2*3, 13, 2*7, 3*5,..) becomes (abs(1-2-3, 2-2-5, 2-3-7, 2-3-3, 2-2-5, 11-2-2, 3-13-2, 7-3-5,..)).

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A143852 (1, 2, 3, 2^2, 5, 2*3, 7, 2^3, 3^2, 2*5, 11, 2^2*3, 13, 2*7, 3*5,..) becomes (abs(1-2-3-2, 2-5-2-3, 7-2-3-3, 2-2-5-11, 2-2-3-13, 2-7-3-5,..)).

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A123948 Triangle read by rows: first row is 1, and n-th row (n > 0) gives the coefficients in the expansion of the characteristic polynomial of the (n - 1)-th Bernstein basis matrix, horizontal flipped.

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A144022 (1, 2, 3, 2^2, 5, 2*3, 7, 2^3, 3^2, 2*5, 11, 2^2*3, 13, 2*7, 3*5,..) becomes (abs(1-2+3-2, 2-5+2-3, 7-2+3-3, 2-2+5-11, 2-2+3-13, 2-7+3-5,..)).

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A144076 (1, 2, 3, 2^2, 5, 2*3, 7, 2^3, 3^2, 2*5, 11, 2^2*3, 13, 2*7, 3*5,..) becomes (abs(1-2*3-2, 2-5*2-3, 7-2*3-3, 2-2*5-11, 2-2*3-13, 2-7*3-5,..)).

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A144114 (1, 2, 3, 2^2, 5, 2*3, 7, 2^3, 3^2, 2*5, 11, 2^2*3, 13, 2*7, 3*5,..) becomes (abs(1-2^3-2, 2-5^2-3, 7-2^3-3, 2-2^5-11, 2-2^3-13, 2-7^3-5,..)).

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A173820 Coefficients of characteristic polynomials of Hadamard Cartan F_2 self-similar 2^n matrices:M={{2, -1}, {-2, 2}}.

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Mathematica

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A143717 (1, 2, 3, 2^2, 5, 23, 7, 2^3, 3^2, 25, 11, 2^23, 13, 27, 3*5,..) becomes (abs(1-2-3, 2-2-5, 2-3-7, 2-3-3, 2-2-5, 11-2-2, 3-13-2, 7-3-5,..)).

A143852 (1, 2, 3, 2^2, 5, 23, 7, 2^3, 3^2, 25, 11, 2^23, 13, 27, 3*5,..) becomes (abs(1-2-3-2, 2-5-2-3, 7-2-3-3, 2-2-5-11, 2-2-3-13, 2-7-3-5,..)).

A144022 (1, 2, 3, 2^2, 5, 23, 7, 2^3, 3^2, 25, 11, 2^23, 13, 27, 3*5,..) becomes (abs(1-2+3-2, 2-5+2-3, 7-2+3-3, 2-2+5-11, 2-2+3-13, 2-7+3-5,..)).

A144076 (1, 2, 3, 2^2, 5, 23, 7, 2^3, 3^2, 25, 11, 2^23, 13, 27, 35,..) becomes (abs(1-23-2, 2-52-3, 7-23-3, 2-25-11, 2-23-13, 2-7*3-5,..)).

A144114 (1, 2, 3, 2^2, 5, 23, 7, 2^3, 3^2, 25, 11, 2^23, 13, 27, 3*5,..) becomes (abs(1-2^3-2, 2-5^2-3, 7-2^3-3, 2-2^5-11, 2-2^3-13, 2-7^3-5,..)).