A201509 -id:A201509 - cp's OEIS frontend

A201780 Riordan array ((1-x)^2/(1-2x), x/(1-2x)).

1, 0, 1, 1, 2, 1, 2, 5, 4, 1, 4, 12, 13, 6, 1, 8, 28, 38, 25, 8, 1, 16, 64, 104, 88, 41, 10, 1, 32, 144, 272, 280, 170, 61, 12, 1, 64, 320, 688, 832, 620, 292, 85, 14, 1, 128, 704, 1696, 2352, 2072, 1204, 462, 113, 16, 1

Offset: 0

Philippe Deléham, Dec 05 2011

Diagonals ascending: 1, 0, 1, 1, 2, 2, 4, 5, 1, 8, 12, 4, ... (see A201509).

			Triangle begins:
  1;
  0,  1;
  1,  2,  1;
  2,  5,  4,  1;
  4, 12, 13,  6,  1;
  8, 28, 38, 25,  8,  1;

Benjamin Braun, W. K. Hough, Matching and Independence Complexes Related to Small Grids, arXiv preprint arXiv:1606.01204 [math.CO], 2016.
Wesley K. Hough, On Independence, Matching, and Homomorphism Complexes, (2017), Theses and Dissertations--Mathematics, 42.

Diagonals: A000012, A005843, A001844, A035597,
Columns: A034008, A045623, A084851, A055585,
Row sums: A052156

CoefficientList[#, y]& /@ CoefficientList[(1-x)^2/(1-(y+2)*x) + O[x]^10, x] // Flatten (* Jean-François Alcover, Nov 03 2018 *)

T(n,k) = 2*T(n-1,k) + T(n-1,k-1) with T(0,0) = 0, T(1,0) = 0, T(2,0) = 0 and T(n,k)= 0 if k < 0 or if n < k.
Sum_{k=0..n} T(n,k)*x^k = A154955(n+1), A034008(n), A052156(n), A055841(n), A055842(n), A055846(n), A055270(n), A055847(n), A055995(n), A055996(n), A056002(n), A056116(n) for x = -1,0,1,2,3,4,5,6,7,8,9,10 respectively.
G.f.: (1-x)^2/(1-(y+2)*x).

A201863 Let CZ(0,x)=1, CZ(1,x)=0 , CZ(2,x)=x^2-1 and CZ(n,x)=2xCZ(n-1,x) - CZ(n-2,x) for n > 2. This sequence is the triangle of polynomial coefficients in order of decreasing exponents.

Original entry on oeis.org

1, 0, 0, 1, 0, -1, 2, 0, -2, 0, 4, 0, -5, 0, 1, 8, 0, -12, 0, 4, 0, 16, 0, -28, 0, 13, 0, -1, 32, 0, -64, 0, 38, 0, -6, 0, 64, 0, -144, 0, 104, 0, -25, 0, 1, 128, 0, -320, 0, 272, 0, -88, 0, 8, 0, 256, 0, -704, 0, 688, 0, -280, 0, 41, 0, -1

Offset: 0

Paul Curtz, Dec 06 2011

From (A039991 without 0's=) A028297 we wrote in A201509
1,  1,
2,  2,
4,  5, 1,
8, 12, 4.
Hence a(n) first coefficients:
1,
0, 0
1, 0,- 1,   x^2-1,
2, 0, -2, 0,
4, 0, -5, 0, 1
8, 0,-12, 0, 4, 0.
The first 1 is a choice.
Row sums=0.
Absolute value row sums: 1 before A163271.
First vertical:A034008=1 before A131577. Third:-A045623.
Mirror image of triangle in A076626. - Philippe Deléham, Dec 07 2011

Cf. A039991, A201509.

A076626 Array of coefficients of polynomials p(n,x) = 2^(n-1)Product_{i=0..n} (x - cos(iPi/n)) of degree (n+1) with P(-1,x) = 1, P(0,x) = 0.

Original entry on oeis.org

1, 0, 0, -1, 0, 1, 0, -2, 0, 2, 1, 0, -5, 0, 4, 0, 4, 0, -12, 0, 8, -1, 0, 13, 0, -28, 0, 16, 0, -6, 0, 38, 0, -64, 0, 32, 1, 0, -25, 0, 104, 0, -144, 0, 64, 0, 8, 0, -88, 0, 272, 0, -320, 0, 128, -1, 0, 41, 0, -280, 0, 688, 0, -704, 0, 256, 0, -10, 0, 170, 0, -832, 0, 1696, 0, -1536, 0, 512, 1, 0, -61, 0, 620, 0, -2352, 0, 4096, 0, -3328, 0, 1024

Offset: 0

Benoit Cloitre, Oct 22 2002

Mirror image of triangle in A201863. - Philippe Deléham, Dec 07 2011

			p(4,x) = 8*x^5 - 12*x^3 + 4*x hence 0,4,0,-12,0,8 are terms in the sequence.
From _Philippe Deléham_, Dec 07 2011: (Start)
Triangle begins:
   1;
   0,    0;
  -1,    0,    1;
   0,   -2,    0,    2;
   1,    0,   -5,    0,    4;
   0,    4,    0,  -12,    0,    8;
  -1,    0,   13,    0,  -28,    0,   16;
   0,   -6,    0,   38,    0,  -64,    0,   32;
   1,    0,  -25,    0,  104,    0, -144,    0,   64; (End)

Cf. A201863, A201509.

T(n,k) = 2*T(n-1,k-1) - T(n-2,k). - Philippe Deléham, Dec 07 2011

Definition corrected by Philippe Deléham, Dec 07 2011

cp's OEIS Frontend

A201780 Riordan array ((1-x)^2/(1-2x), x/(1-2x)).

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A201863 Let CZ(0,x)=1, CZ(1,x)=0 , CZ(2,x)=x^2-1 and CZ(n,x)=2xCZ(n-1,x) - CZ(n-2,x) for n > 2. This sequence is the triangle of polynomial coefficients in order of decreasing exponents.

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A076626 Array of coefficients of polynomials p(n,x) = 2^(n-1)Product_{i=0..n} (x - cos(iPi/n)) of degree (n+1) with P(-1,x) = 1, P(0,x) = 0.

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cp's OEIS Frontend

A201780 Riordan array ((1-x)^2/(1-2x), x/(1-2x)).

Views

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Comments

Examples

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Mathematica

Formula

A201863 Let CZ(0,x)=1, CZ(1,x)=0 , CZ(2,x)=x^2-1 and CZ(n,x)=2*x*CZ(n-1,x) - CZ(n-2,x) for n > 2. This sequence is the triangle of polynomial coefficients in order of decreasing exponents.

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A076626 Array of coefficients of polynomials p(n,x) = 2^(n-1)*Product_{i=0..n} (x - cos(i*Pi/n)) of degree (n+1) with P(-1,x) = 1, P(0,x) = 0.

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A201863 Let CZ(0,x)=1, CZ(1,x)=0 , CZ(2,x)=x^2-1 and CZ(n,x)=2xCZ(n-1,x) - CZ(n-2,x) for n > 2. This sequence is the triangle of polynomial coefficients in order of decreasing exponents.

A076626 Array of coefficients of polynomials p(n,x) = 2^(n-1)Product_{i=0..n} (x - cos(iPi/n)) of degree (n+1) with P(-1,x) = 1, P(0,x) = 0.