A133156 -id:A133156 - cp's OEIS frontend

A140882 Triangle by rows with row n formed by coefficients of the characteristic polynomial of the n X n tridiagonal matrix with m_{i,i} = 2 for i=1..n, m_{i,i-1} = m_{i,i+1} = -1 for i=2..n-1, and m_{1,2} = m_{n,n-1} = -2.

1, 2, -1, 0, -4, 1, 0, -8, 6, -1, 0, -12, 19, -8, 1, 0, -16, 44, -34, 10, -1, 0, -20, 85, -104, 53, -12, 1, 0, -24, 146, -259, 200, -76, 14, -1, 0, -28, 231, -560, 606, -340, 103, -16, 1, 0, -32, 344, -1092, 1572, -1210, 532, -134, 18, -1, 0, -36, 489, -1968, 3630, -3652, 2171, -784, 169, -20, 1

Offset: 0

Roger L. Bagula and Gary W. Adamson, Jul 22 2008

			  1;
  2,  -1;
  0,  -4,   1;
  0,  -8,   6,    -1;
  0, -12,  19,    -8,    1;
  0, -16,  44,   -34,   10,    -1;
  0, -20,  85,  -104,   53,   -12,    1;
  0, -24, 146,  -259,  200,   -76,   14,   -1;
  0, -28, 231,  -560,  606,  -340,  103,  -16,   1;
  0, -32, 344, -1092, 1572, -1210,  532, -134,  18,  -1;
  0, -36, 489, -1968, 3630, -3652, 2171, -784, 169, -20, 1;
  ...

Kemeny, Snell and Thompson, Introduction to Finite Mathematics, 1966, Prentice-Hall, New Jersey, Section 3, Chapter VII, page 407.

Pentti Haukkanen, Jorma Merikoski, and Seppo Mustonen, Some polynomials associated with regular polygons, Acta Univ. Sapientiae, Mathematica, 6, 2 (2014) 178-193.

Cf. A228785, A267633.

Cf. A049310, A133156.

# Assume T(1, 0) = 0 instead of 2.
# Then a slightly modified form of Copeland's second comment gives
# T(n, k) = [t^k] [x^n] gf where
gf := ((t - 4)*t*x^2) / ((t - 2)*x + x^2 + 1) - t*x + 1:
ser := series(gf, x, 12): cx := n -> coeff(ser, x, n):
for n from 0 to 10 do lprint(seq(coeff(cx(n), t, k), k = 0..n)) od;
# Peter Luschny, Apr 27 2024

T[n_, m_, d_] := If[ n == m, 2, If[(n == d && m == d - 1) || ( n == 1 && m == 2), -2, If[(n == m - 1 || n == m + 1), -1, 0]]];
M[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}];
a = Join[{{1}}, Table[CoefficientList[Det[M[ d] - x*IdentityMatrix[d]], x], {d, 1, 10}]];
Flatten[a]

Starting with the third row (0, -4, 1), these coefficients appear to occur in the odd row polynomials in inverse powers of u for the o.g.f. (u - 4)/(u - u*x + x^2) of A267633, which generates a Fibonacci-type running average of adjacent polynomials of the o.g.f. involving these polynomials and others embedded in A228785. - Tom Copeland, Jan 16 2016

A generating function for the third and later shifted, unsigned rows is (4 + t) / (1 - (2 + t)*x + x^2) = Sum_{n >= 0} (4+t)*U_n((2 + t)/2)*x^n = 4 + t + (8 + 6*t + t^2)*x + ..., where U_n(t) are the Chebyshev polynomials of the second kind of A133156. U_n((2 + t)/2) = V_n((2 + t)), where V_n(t) are the Chebyshev polynomials of A049310. A formula for the coefficients is given by Eqn. 8 in Haukkanen et al. - Tom Copeland, Apr 26 2024

Edited by the editors of the OEIS, Apr 27 2024

A317506 Triangle read by rows: T(0,0) = 1; T(n,k) = 2 T(n-1,k) - T(n-4,k-1) for 0 <= k <= floor(n/4); T(n,k)=0 for n or k < 0.

Original entry on oeis.org

1, 2, 4, 8, 16, -1, 32, -4, 64, -12, 128, -32, 256, -80, 1, 512, -192, 6, 1024, -448, 24, 2048, -1024, 80, 4096, -2304, 240, -1, 8192, -5120, 672, -8, 16384, -11264, 1792, -40, 32768, -24576, 4608, -160, 65536, -53248, 11520, -560, 1, 131072, -114688, 28160, -1792, 10

Offset: 0

Shara Lalo, Aug 31 2018

The numbers in rows of the triangle are along "third layer" skew diagonals pointing top-right in center-justified triangle given in A065109 ((2-x)^n) and along "third layer" skew diagonals pointing top-left in center-justified triangle given in A303872 ((-1+2x)^n), see links. (Note: First layer skew diagonals in center-justified triangles of coefficients in expansions of (2-x)^n and (-1+2x)^n are given in A133156 (coefficients of Chebyshev polynomials of the second kind) and A305098 respectively.) The coefficients in the expansion of 1/(1-2x+x^4) are given by the sequence generated by the row sums. The row sums give A008937. If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 1.83928675521416113... (A058265: Decimal expansion of the tribonacci constant t, the real root of x^3-x^2-x-1), when n approaches infinity.

			Triangle begins:
       1;
       2;
       4;
       8;
      16,      -1;
      32,      -4;
      64,     -12;
     128,     -32;
     256,     -80,     1;
     512,    -192,     6;
    1024,    -448,    24;
    2048,   -1024,    80;
    4096,   -2304,   240,    -1;
    8192,   -5120,   672,    -8;
   16384,  -11264,  1792,   -40;
   32768,  -24576,  4608,  -160;
   65536,  -53248, 11520,  -560,  1;
  131072, -114688, 28160, -1792, 10;
  262144, -245760, 67584, -5376, 60;

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3.

Row sums give A008937.
Cf. A065109, A303872.
Cf. A133156, A305098.
Cf. A058265.

t[n_, k_] := t[n, k] = 2^(n - 4 k) * (-1)^k/((n - 4 k)! k!) * (n - 3 k)!; Table[t[n, k], {n, 0, 16}, {k, 0, Floor[n/4]} ]  // Flatten
t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, 2 * t[n - 1, k] - t[n - 4, k - 1]]; Table[t[n, k], {n, 0, 16}, {k, 0, Floor[n/4]}] // Flatten

T(n,k) = 2^(n - 4*k) * (-1)^k / ((n - 4*k)! k!) * (n - 3*k)! where n >= 0 and 0 <= k <= floor(n/4).

A185342 Triangle of successive recurrences in columns of A117317(n).

Original entry on oeis.org

2, 4, -4, 6, -12, 8, 8, -24, 32, -16, 10, -40, 80, -80, 32, 12, -60, 160, -240, 192, -64, 14, -84, 280, -560, 672, -448, 128, 16, -112, 448, -1120, 1792, -1792, 1024, -256, 18, -144, 672, -2016, 4032, -5376, 4608, -2304, 512, 20, -180, 960, -3360, 8064

Offset: 0

Paul Curtz, Jan 26 2012

A117317 (A):
1
2    1
4    5   1
8   16   9   1
16  44  41  14   1
32 112 146  85  20  1
64 272 456 377 155 27 1
have for their columns successive signatures
(2) (4,-4) (6,-12,8) (8,-24, 32, -16) (10,-40,80,-80,32) i.e. a(n).
Take based on abs(A133156) (B):
1
2   0
4   1  0
8   4  0 0
16 12  1 0 0
32 32  6 0 0 0
64 80 24 1 0 0 0.
The recurrences of successive columns are also a(n). a(n) columns: A005843(n+1), A046092(n+1), A130809, A130810, A130811, A130812, A130813.
A053220 + A001787 = A014480.

			Triangle T(n,k),for 1<=k<=n, begins :
2                                         (1)
4    -4                                   (2)
6   -12   8                               (3)
8   -24  32   -16                         (4)
10  -40  80   -80   32                    (5)
12  -60 160  -240  192   -64              (6)
14  -84 280  -560  672  -448  128         (7)
16 -112 448 -1120 1792 -1792 1024 -256    (8)
Successive rows can be divided by A171977.

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Cf. For (A): A053220, A056243. For (B): A000079, A001787, A001788, A001789. For A193862: A115068 (a Coxeter group). For (2): A014480 (also (3),(4),(5),..); also A053220 and A001787.

Cf. A007318.

Table[(-1)*Binomial[n, k]*(-2)^k, {n, 1, 20}, {k, 1, n}] // Flatten (* G. C. Greubel, Jun 27 2017 *)

for(n=1,20, for(k=1,n, print1((-2)^(k+1)*binomial(n,k)/2, ", "))) \\ G. C. Greubel, Jun 27 2017

Take A133156(n) without 1's or -1's double triangle (C)=
2
4
8      -4
16    -12
32    -32   6
64    -80  24
128  -192  80   -8
256  -448 240  -40
512 -1024 672 -160 10;
a(n) is increasing odd diagonals and increasing (sign changed) even diagonals. Rows sum of (C) = A201629 (?) Another link between Chebyshev polynomials and cos( ).
Absolute values: A013609(n) without 1's. Also 2*A193862 = (2*A002260)*A135278.
T(n,k) = T(n-1,k) - 2*T(n-1,k-1) for k>1, T(n,1) = 2*n = 2*T(n-1,1) - T(n-2,1). - Philippe Deléham, Feb 11 2012
T(n,k) = (-1)* Binomial(n,k)*(-2)^k, 1<=k<=n. - Philippe Deléham, Feb 11 2012

A228637 The number triangle associated with the polynomials V_n(x).

Original entry on oeis.org

1, -1, 1, -1, 1, 1, 1, 1, 3, 1, 1, 1, 11, 5, 1, -1, 1, 41, 29, 7, 1, -1, 1, 153, 169, 55, 9, 1, 1, 1, 571, 985, 433, 89, 11, 1, 1, 1, 2131, 5741, 3409, 881, 131, 13, 1, -1, 1, 7953, 33461, 26839, 8721, 1561, 181, 15, 1

Offset: 0

Jonny Griffiths, Aug 28 2013

V(n) is the polynomial with integer coefficients in x given by cos((2n+1)(arccos(x)/2))/(arccos(x)/2). The triangle here is given by V_0(0), V_1(0), V_0(1), V_2(0), V_1(1), V_0(2), V_3(0), V_2(1), V_1(2), V_0(3), V_4(0),....

			V_0(x)=1, V_1(x)=2x-1, V_2(x)=4x^2-2x-1,  ...

Cf. A028297, A101124, A133156, A228161, A228565, A180870.

The terms are given by the recurrence relation V_{n+1}(x) = 2xV_n(x)-V_{n-1}(x), V_0(x) = 1, V_1(x)=2x-1.

A317504 Triangle read by rows: T(0,0) = 1; T(n,k) = 2 T(n-1,k) - T(n-3,k-1) for k = 0..floor(n/3); T(n,k)=0 for n or k < 0.

Original entry on oeis.org

1, 2, 4, 8, -1, 16, -4, 32, -12, 64, -32, 1, 128, -80, 6, 256, -192, 24, 512, -448, 80, -1, 1024, -1024, 240, -8, 2048, -2304, 672, -40, 4096, -5120, 1792, -160, 1, 8192, -11264, 4608, -560, 10, 16384, -24576, 11520, -1792, 60, 32768, -53248, 28160, -5376, 280, -1, 65536, -114688, 67584, -15360, 1120, -12

Offset: 0

Shara Lalo, Aug 02 2018

The numbers in rows of the triangle are along "second layer" skew diagonals pointing top-right in center-justified triangle given in A065109 ((2-x)^n) and along "second layer" skew diagonals pointing top-left in center-justified triangle given in A303872 ((-1+2x)^n), see links. (Note: First layer skew diagonals in center-justified triangles of coefficients in expansions of (2-x)^n and (-1+2x)^n are given in A133156 (coefficients of Chebyshev polynomials of the second kind) and A305098 respectively.) The coefficients in the expansion of 1/(1-2x+x^3) are given by the sequence generated by the row sums. The row sums give A000071 (Fibonacci numbers - 1). If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 1.61803398874989484... (A001622: Decimal expansion of Golden ratio (phi or tau) = (1 + sqrt(5))/2), when n approaches infinity.

			Triangle begins:
       1;
       2;
       4;
       8,      -1;
      16,      -4;
      32,     -12;
      64,     -32,      1;
     128,     -80,      6;
     256,    -192,     24;
     512,    -448,     80,      -1;
    1024,   -1024,    240,      -8;
    2048,   -2304,    672,     -40;
    4096,   -5120,   1792,    -160,     1;
    8192,  -11264,   4608,    -560,    10;
   16384,  -24576,  11520,   -1792,    60;
   32768,  -53248,  28160,   -5376,   280,   -1;
   65536, -114688,  67584,  -15360,  1120,  -12;
  131072, -245760, 159744,  -42240,  4032,  -84;
  262144, -524288, 372736, -112640, 13440, -448, 1;

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 139-141, 391-393.

Row sums give A000071.
Cf. A065109, A303872.
Cf. A133156, A305098.
Cf. A001622.

t[n_, k_] := t[n, k] = 2^(n - 3k) * (-1)^k/((n - 3 k)! k!) * (n - 2 k)!; Table[t[n, k], {n, 0, 16}, {k, 0, Floor[n/3]} ]  // Flatten
t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, 2 * t[n - 1, k] - t[n - 3, k - 1]]; Table[t[n, k], {n, 0, 16}, {k, 0, Floor[n/3]}] // Flatten

T(n,k) = 2^(n - 3k) * (-2)^k / ((n - 3k)! k!) * (n - 2k)! where n is a nonnegative integer and k = 0..floor(n/3).

A317505 Triangle read by rows: T(0,0) = 1; T(n,k) = - T(n-1,k) - 2 T(n-3,k-1) for k = 0..floor(n/3); T(n,k)=0 for n or k < 0.

Original entry on oeis.org

1, -1, 1, -1, 2, 1, -4, -1, 6, 1, -8, 4, -1, 10, -12, 1, -12, 24, -1, 14, -40, 8, 1, -16, 60, -32, -1, 18, -84, 80, 1, -20, 112, -160, 16, -1, 22, -144, 280, -80, 1, -24, 180, -448, 240, -1, 26, -220, 672, -560, 32, 1, -28, 264, -960, 1120, -192, -1, 30, -312, 1320, -2016, 672, 1, -32, 364, -1760, 3360, -1792, 64, -1, 34, -420, 2288, -5280, 4032, -448

Offset: 0

Shara Lalo, Aug 02 2018

The numbers in rows of the triangle are along "second layer" skew diagonals pointing top-left in center-justified triangle given in A065109 ((2-x)^n) and along "second layer" skew diagonals pointing top-right in center-justified triangle given in A303872 ((-1+2x)^n), see links. (Note: First layer skew diagonals in center-justified triangles of coefficients in expansions of (2-x)^n and (-1+2x)^n are given in A133156 (coefficients of Chebyshev polynomials of the second kind) and A305098 respectively.) The coefficients in the expansion of 1/(1+x+2x^3) are given by the sequence generated by the row sums (see A077973).

			Triangle begins:
   1;
  -1;
   1;
  -1,   2;
   1,  -4;
  -1,   6;
   1,  -8,    4;
  -1,  10,  -12;
   1, -12,   24;
  -1,  14,  -40,     8;
   1, -16,   60,   -32;
  -1,  18,  -84,    80;
   1, -20,  112,  -160,    16;
  -1,  22, -144,   280,   -80;
   1, -24,  180,  -448,   240;
  -1,  26, -220,   672,  -560,    32;
   1, -28,  264,  -960,  1120,  -192;
  -1,  30, -312,  1320, -2016,   672;
   1, -32,  364, -1760,  3360, -1792,   64;
  -1,  34, -420,  2288, -5280,  4032, -448;

Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 139-141, 391-393.

Row sums give A077973.
Cf. A065109, A303872.
Cf. A133156, A305098.

t[n_, k_] := t[n, k] = (-1)^(n - 3k) * 2^k/((n - 3 k)! k!) * (n - 2 k)!; Table[t[n, k], {n, 0, 19}, {k, 0, Floor[n/3]} ]  // Flatten
t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, - t[n - 1, k] + 2 t[n - 3, k - 1]]; Table[t[n, k], {n, 0, 19}, {k, 0, Floor[n/3]}] // Flatten

T(n,k) = (-1)^(n - 3k) * 2^k / ((n - 3k)! k!) * (n - 2k)! where n is a nonnegative integer and k = 0..floor(n/3).

A228356 The triangle associated with the family of polynomials W_n(x).

Original entry on oeis.org

1, 1, 1, -1, 3, 1, -1, 5, 5, 1, 1, 7, 19, 7, 1, 1, 9, 71, 41, 9, 1, -1, 11, 265, 239, 71, 11, 1, -1, 13, 989, 1393, 559, 109, 13, 1, 1, 15, 3691, 8119, 4401, 1079, 155, 15, 1, 1, 17, 13775, 47321, 34649, 10681, 1847, 209, 17, 1

Offset: 0

Jonny Griffiths, Aug 28 2013

W_n(x) is the family of polynomials in x with integer coefficients given by W_n(x) = sin((2n+1)arccos(x)/2)/(sin(arccos(x)/2)).

These polynomials are intimately linked with the Chebyshev polynomials of the first and second kinds, and represent the polynomials associated with the Dirichlet kernel.

			The triangle is given here as W_0(0)=1, W_1(0)=1, W_0(1)=1, W_2(0)=-1, W_1(1)=3, W_0(2)=1, W_3(0)=-1, W_2(1)=5 ...

Cf. A028297, A101124, A133156, A228161, A228565, A228637, A180870.

W[0, ] = 1; W[1, x] := 2 x + 1; W[n_, x_] := W[n, x] = 2 x W[n - 1, x] - W[n - 2, x]; Table[W[n - x, x] , {n, 0, 9}, {x, 0, n}] // Flatten (* Jean-François Alcover, Jun 11 2017 *)

W_{n+1} = 2xW_n(x) - W_{n-1}, W_0(x)=1, W_1(x)=2x+1.

cp's OEIS Frontend

A140882 Triangle by rows with row n formed by coefficients of the characteristic polynomial of the n X n tridiagonal matrix with m_{i,i} = 2 for i=1..n, m_{i,i-1} = m_{i,i+1} = -1 for i=2..n-1, and m_{1,2} = m_{n,n-1} = -2.

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A317506 Triangle read by rows: T(0,0) = 1; T(n,k) = 2 T(n-1,k) - T(n-4,k-1) for 0 <= k <= floor(n/4); T(n,k)=0 for n or k < 0.

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A185342 Triangle of successive recurrences in columns of A117317(n).

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A228637 The number triangle associated with the polynomials V_n(x).

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A317504 Triangle read by rows: T(0,0) = 1; T(n,k) = 2 T(n-1,k) - T(n-3,k-1) for k = 0..floor(n/3); T(n,k)=0 for n or k < 0.

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A317505 Triangle read by rows: T(0,0) = 1; T(n,k) = - T(n-1,k) - 2 T(n-3,k-1) for k = 0..floor(n/3); T(n,k)=0 for n or k < 0.

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A228356 The triangle associated with the family of polynomials W_n(x).

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