A008312 -id:A008312 - cp's OEIS frontend

A008310 Triangle of coefficients of Chebyshev polynomials T_n(x).

1, 1, -1, 2, -3, 4, 1, -8, 8, 5, -20, 16, -1, 18, -48, 32, -7, 56, -112, 64, 1, -32, 160, -256, 128, 9, -120, 432, -576, 256, -1, 50, -400, 1120, -1280, 512, -11, 220, -1232, 2816, -2816, 1024, 1, -72, 840, -3584, 6912, -6144, 2048, 13, -364, 2912, -9984, 16640, -13312, 4096

Offset: 0

N. J. A. Sloane

The row length sequence of this irregular array is A008619(n), n >= 0. Even or odd powers appear in increasing order starting with 1 or x for even or odd row numbers n, respectively. This is the standard triangle A053120 with 0 deleted. - Wolfdieter Lang, Aug 02 2014

Let T* denote the triangle obtained by replacing each number in this triangle by its absolute value. Then T* gives the coefficients for cos(nx) as a polynomial in cos x. - Clark Kimberling, Aug 04 2024

			Rows are: (1), (1), (-1,2), (-3,4), (1,-8,8), (5,-20,16) etc., since if c = cos(x): cos(0x) = 1, cos(1x) = 1c; cos(2x) = -1+2c^2; cos(3x) = -3c+4c^3, cos(4x) = 1-8c^2+8c^4, cos(5x) = 5c-20c^3+16c^5, etc.
From _Wolfdieter Lang_, Aug 02 2014: (Start)
This irregular triangle a(n,k) begins:
  n\k   0    1     2      3      4      5      6      7 ...
  0:    1
  1:    1
  2:   -1    2
  3:   -3    4
  4:    1   -8     8
  5:    5  -20    16
  6:   -1   18   -48     32
  7:   -7   56  -112     64
  8:    1  -32   160   -256    128
  9:    9 -120   432   -576    256
 10:   -1   50  -400   1120  -1280    512
 11:  -11  220 -1232   2816  -2816   1024
 12:    1  -72   840  -3584   6912  -6144   2048
 13:   13 -364  2912  -9984  16640 -13312   4096
 14:   -1   98 -1568   9408 -26880  39424 -28672   8192
 15:  -15  560 -6048  28800 -70400  92160 -61440  16384
  ...
T(4,x) = 1 - 8*x^2 + 8*x^4, T(5,x) = 5*x - 20*x^3 +16*x^5.
(End)

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 795.
E. A. Guilleman, Synthesis of Passive Networks, Wiley, 1957, p. 593.
Yaroslav Zolotaryuk, J. Chris Eilbeck, "Analytical approach to the Davydov-Scott theory with on-site potential", Physical Review B 63, p543402, Jan. 2001. The authors write, "Since the algebra of these is 'hyperbolic', contrary to the usual Chebyshev polynomials defined on the interval 0 <= x <= 1, we call the set of functions (21) the hyperbolic Chebyshev polynomials." (This refers to the triangle T* described in Comments.)

R. J. Mathar, Table of n, a(n) for n = 0..2600
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Renato Ferreira Pinto Jr. and Nathaniel Harms, Testing Support Size More Efficiently Than Learning Histograms, arXiv:2410.18915 [cs.DS], 2024. See p. 40.
D. Foata and G.-N. Han, Nombres de Fibonacci et polynomes orthogonaux
C. Lanczos, Applied Analysis (Annotated scans of selected pages)
I. Rivin, Growth in free groups (and other stories), arXiv:math/9911076 [math.CO], 1999.
Eric Weisstein's World of Mathematics, Chebyshev Polynomial of the First Kind.
Wikipedia, Chebyshev polynomials.
Index entries for sequences related to Chebyshev polynomials.

A039991 is a row reversed version, but has zeros which enable the triangle to be seen. Columns/diagonals are A011782, A001792, A001793, A001794, A006974, A006975, A006976 etc.
Reflection of A028297. Cf. A008312, A053112.
Row sums are one. Polynomial evaluations include A001075 (x=2), A001541 (x=3), A001091, A001079, A023038, A011943, A001081, A023039, A001085, A077422, A077424, A097308, A097310, A068203.
Cf. A053120.

A008310 := proc(n,m) local x ; coeftayl(simplify(ChebyshevT(n,x),'ChebyshevT'),x=0,m) ; end: i := 0 : for n from 0 to 100 do for m from n mod 2 to n by 2 do printf("%d %d ",i,A008310(n,m)) ; i := i+1 ; od ; od ; # R. J. Mathar, Apr 20 2007
# second Maple program:
b:= proc(n) b(n):= `if`(n<2, 1, expand(2*b(n-1)-x*b(n-2))) end:
T:= n-> (p-> (d-> seq(coeff(p, x, d-i), i=0..d))(degree(p)))(b(n)):
seq(T(n), n=0..15);  # Alois P. Heinz, Sep 04 2019

Flatten[{1, Table[CoefficientList[ChebyshevT[n, x], x], {n, 1, 13}]}]//DeleteCases[#, 0, Infinity]& (* or *) Flatten[{1, Table[Table[((-1)^k*2^(n-2 k-1)*n*Binomial[n-k, k])/(n-k), {k, Floor[n/2], 0, -1}], {n, 1, 13}]}] (* Eugeniy Sokol, Sep 04 2019 *)

a(n,m) = 2^(m-1) * n * (-1)^((n-m)/2) * ((n+m)/2-1)! / (((n-m)/2)! * m!) if n>0. - R. J. Mathar, Apr 20 2007
From Paul Weisenhorn, Oct 02 2019: (Start)
T_n(x) = 2*x*T_(n-1)(x) - T_(n-2)(x), T_0(x) = 1, T_1(x) = x.
T_n(x) = ((x+sqrt(x^2-1))^n + (x-sqrt(x^2-1))^n)/2. (End)
From Peter Bala, Aug 15 2022: (Start)
T(n,x) = [z^n] ( z*x + sqrt(1 + z^2*(x^2 - 1)) )^n.
Sum_{k = 0..2*n} binomial(2*n,k)*T(k,x) = (2^n)*(1 + x)^n*T(n,x).
exp( Sum_{n >= 1} T(n,x)*t^n/n ) = Sum_{n >= 0} P(n,x)*t^n, where P(n,x) denotes the n-th Legendre polynomial. (End)

Additional comments and more terms from Henry Bottomley, Dec 13 2000

Edited: Corrected Cf. A039991 statement. Cf. A053120 added. - Wolfdieter Lang, Aug 06 2014

A099089 Riordan array (1, 2+x).

Original entry on oeis.org

1, 0, 2, 0, 1, 4, 0, 0, 4, 8, 0, 0, 1, 12, 16, 0, 0, 0, 6, 32, 32, 0, 0, 0, 1, 24, 80, 64, 0, 0, 0, 0, 8, 80, 192, 128, 0, 0, 0, 0, 1, 40, 240, 448, 256, 0, 0, 0, 0, 0, 10, 160, 672, 1024, 512, 0, 0, 0, 0, 0, 1, 60, 560, 1792, 2304, 1024, 0, 0, 0, 0, 0, 0, 12, 280, 1792, 4608, 5120, 2048

Offset: 0

Paul Barry, Sep 25 2004

Row sums are A000129. Diagonal sums are A008346. The Riordan array (1, s+tx) defines T(n,k) = binomial(k,n-k)*s^k*(t/s)^(n-k). The row sums satisfy a(n) = s*a(n-1) + t*a(n-2) and the diagonal sums satisfy a(n) = s*a(n-2) + t*a(n-3).
Triangle T(n,k), 0 <= k <= n, read by rows given by [0, 1/2, -1/2, 0, 0, 0, 0, ...] DELTA [2, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 10 2008
As an upper right triangle (in the example), table rows give number of points, edges, faces, cubes, 4D hypercubes etc. in hypercubes of increasing dimension by column. - Henry Bottomley, Apr 14 2000. More precisely, the (i,j)-th entry is the number of j-dimensional subspaces of an i-dimensional hypercube (see the Coxeter reference). - Christof Weber, May 08 2009

			Triangle begins:
  1;
  0,  2;
  0,  1,  4;
  0,  0,  4,  8;
  0,  0,  1, 12, 16;
  0,  0,  0,  6, 32, 32;
  0,  0,  0,  1, 24, 80, 64;
The entries can also be interpreted as the antidiagonal reading of the following array:
  1,    2,    4,    8,   16,   32,   64,  128,  256,  512, 1024,... A000079
  0,    1,    4,   12,   32,   80,  192,  448, 1024, 2304, 5120,... A001787
  0,    0,    1,    6,   24,   80,  240,  672, 1792, 4608,11520,... A001788
  0,    0,    0,    1,    8,   40,  160,  560, 1792, 5376,15360,... A001789
  0,    0,    0,    0,    1,   10,   60,  280, 1120, 4032,13440,...
  0,    0,    0,    0,    0,    1,   12,   84,  448, 2016, 8064,...
  0,    0,    0,    0,    0,    0,    1,   14,  112,  672, 3360,...
  0,    0,    0,    0,    0,    0,    0,    1,   16,  144,  960,...
  0,    0,    0,    0,    0,    0,    0,    0,    1,   18,  180,...
  0,    0,    0,    0,    0,    0,    0,    0,    0,    1,   20,...
  0,    0,    0,    0,    0,    0,    0,    0,    0,    0,    1,...

H. S. M. Coxeter, Regular Polytopes, Dover Publications, New York (1973), p. 122.

Eric W. Weisstein's Mathworld, Hypercube.

Cf. A053118, A008312, A062715, A038207.

Number triangle T(n,k) = binomial(k, n-k)*2^k*(1/2)^(n-k); columns have g.f. (2*x+x^2)^k.
G.f.: 1/(1-2y*x-y*x^2). - Philippe Deléham, Nov 20 2011
Sum_ {k=0..n} T(n,k)*x^k = A000007(n), A000129(n+1), A090017(n+1), A090018(n), A190510(n+1), A190955(n+1) for x = 0,1,2,3,4,5 respectively. - Philippe Deléham, Nov 20 2011
T(n,k) = 2*T(n-1,k-1) + T(n-2,k-1), T(0,0) = 1, T(1,0) = T(2,0) = 0, T(1,1) = 2, T(2,1) = 1, T(2,2) = 4, T(n,k) = 0 if k > n or if k < 0. - Philippe Deléham, Oct 30 2013

A180870 D(n, x) is the Dirichlet kernel sin((n+1/2)x)/sin(x/2). The triangle gives in row n the coefficients of descending powers of x of the polynomial D(n, arccos(x)).

Original entry on oeis.org

1, 2, 1, 4, 2, -1, 8, 4, -4, -1, 16, 8, -12, -4, 1, 32, 16, -32, -12, 6, 1, 64, 32, -80, -32, 24, 6, -1, 128, 64, -192, -80, 80, 24, -8, -1, 256, 128, -448, -192, 240, 80, -40, -8, 1, 512, 256, -1024, -448, 672, 240, -160, -40, 10, 1

Offset: 0

Jonny Griffiths, Sep 21 2010

D(n, arccos(x)) = U(n, x) + U(n-1, x) where U(n, x) are the Chebyshev polynomials of the second kind. These polynomials arise naturally in the investigation of the integer triples (p, q, (p*q + 1)/(p + q)).

Chebyshev polynomials of the fourth kind, usually denoted by W(n, x) (see, for example, Mason and Handscomb, Chapter 1, Definition 1.3). See A228565 for Chebyshev polynomials of the third kind. Cf. A157751. - Peter Bala, Jan 17 2014

			The triangle T(n,m) begins:
n\m    0   1     2     3    4   5    6    7   8  9  10 ...
0:     1
1:     2   1
2:     4   2    -1
3:     8   4    -4    -1
4:    16   8   -12    -4    1
5:    32  16   -32   -12    6   1
6:    64  32   -80   -32   24   6   -1
7:   128  64  -192   -80   80  24   -8   -1
8:   256 128  -448  -192  240  80  -40   -8   1
9:   512 256 -1024  -448  672 240 -160  -40  10  1
10: 1024 512 -2304 -1024 1792 672 -560 -160  60 10  -1
... reformatted - _Wolfdieter Lang_, Jul 26 2014
Recurrence: T(4,2) = (1 + 1)*T(3,2) - T(3,1) = 2*(-4) - 4 = -12. T(4,3) = 0*T(3,3) - (-1)*T(3,2) = T(3,2) = -4. - _Wolfdieter Lang_, Jul 30 2014

J. C. Mason and D. C. Handscomb, Chebyshev polynomials, Chapman and Hall/CRC 2002.

Paul Barry, On the Group of Almost-Riordan Arrays, arXiv preprint arXiv:1606.05077 [math.CO], 2016.
Wikipedia, Dirichlet kernel.

Cf. A008312, A028297, A157751, A228565, A049310, A244419 (row reversed triangle).

ogf := (1 + t)/(1 - 2*x*t + t^2):
ser := simplify(series(ogf, t, 12)): tc := n -> coeff(ser, t, n):
Trow := n -> local k; seq(coeff(tc(n), x, n-k), k = 0..n):
seq(print(Trow(n)), n = 0..9);  # Peter Luschny, Oct 07 2024

row(n) = {if (n==0, return([1])); f = 2*x+1; for (k = 2, n, for (i = 1, (k-1)\2 + 1, f += (-1)^(i+1)*(binomial(k-i, i-1)*(2*x)^(k-2*i+2) - 2*binomial(k-1-i, i-1)*(2*x)^(k-2*i)););); Vec(f);} \\ Michel Marcus, Jul 18 2014

From Peter Bala, Jan 17 2014: (Start)
O.g.f. (1 + t)/(1 - 2*x*t + t^2) = 1 + (2*x + 1)*t + (4*x^2 + 2*x - 1)*t^2 + ....
Recurrence equation: W(0,x) = 1, W(1,x) = 2*x + 1 and W(n,x) = 2*x*W(n-1,x) - W(n-2,x) for n >= 2.
In terms of U(n,x), the Chebyshev polynomials of the second kind, we have W(n,x) = U(2*n,u) with u = sqrt((1 + x)/2). Also binomial(2*n,n)*W(n,x) = 2^(2*n)*Jacobi_P(n,1/2,-1/2,x). (End)
Row sums: 2*n+1. - Michel Marcus, Jul 16 2014
T(n,m) = [x^(n-m)](U(n, x) + U(n-1, x)) = [x^(n-m)] S(2*n, sqrt(2*(1+x))), n >= m >= 0, with U(n, x) = S(n, 2*x). The coefficient triangle of the Chebyshev S-polynomials is given in A049310. See the Peter Bala comments above. - Wolfdieter Lang, Jul 26 2014
From Wolfdieter Lang, Jul 30 2014: (Start)
O.g.f. for the row polynomials R(n,x) = Sum_{m=0..n} T(n,m)*x^m, obtained from the one given by Peter Bala above by row reversion: (1 + x*t)/(1 - 2*t + (x*t)^2).
In analogy to A157751 one can derive a recurrence for the row polynomials R(n, x) = x^n*Dir(n,1/x) with Dir(n,x) = U(n,x) + U(n-1,x) using also negative arguments but only one recursive step: R(n,x) = (1+x)*R(n-1,-x) + R(n-1,x), n >= 1, R(0,x) = 1 (R(-1,x) = -1/x). Proof: derive the o.g.f. and compare it with the known one.
This entails the triangle recurrence T(n,m) = (1 + (-1)^m)* T(n-1,m) - (-1)^m*T(n-1,m-1), for n >= m >= 1 with T(n,m) = 0 if n < m and T(n,0) = 2^n. (End)

Missing term in sequence corrected by Paul Curtz, Dec 31 2011

Edited (name reformulated, Wikipedia link added) by Wolfdieter Lang, Jul 26 2014

A228565 Triangle read by rows: coefficients of descending powers of the polynomial V(n,x) = cos((2n+1)(arccos(x)/2))/cos(arccos(x)/2), n >= 0.

Original entry on oeis.org

1, 2, -1, 4, -2, -1, 8, -4, -4, 1, 16, -8, -12, 4, 1, 32, -16, -32, 12, 6, -1, 64, -32, -80, 32, 24, -6, -1, 128, -64, -192, 80, 80, -24, -8, 1, 256, -128, -448, 192, 240, -80, -40, 8, 1, 512, -256, -1024, 448, 672, -240, -160, 40, 10, -1, 1024, -512, -2304, 1024, 1792, -672, -560, 160, 60, -10, -1, 2048, -1024, -5120, 2304, 4608, -1792, -1792, 560, 280, -60, -12, 1, 4096, -2048, -11264, 5120, 11520, -4608, -5376, 1792, 1120, -280, -84, 12, 1

Offset: 0

Jonny Griffiths, Aug 25 2013

V(n,x) is related to the Dirichlet kernel and its associated polynomials. V(n,x) arises in studying recurrences connecting the Chebyshev polynomials of the first and second kinds. It differs from A180870 above only in the signs of terms.

Chebyshev polynomials V(n,x) of the third kind (see, for example, Mason and Handscomb, Chapter 1, Definition 1.3). See A180870 for Chebyshev polynomials of the fourth kind. Cf. A155751. - Peter Bala, Jan 17 2014

			V(0,x) = 1, V(1,x) = 2x-1, V(2,x) = 4x^2-2x-1, V(3,x) = 8x^3 -4x^2 - 4x + 1, V(4,x) = 16x^4 - 8x^3 - 12x^2 + 4x + 1, V(5,x) = 32x^5 - 16x^4 - 32x^3 + 12x^2 + 6x - 1, V(6,x) =64x^6 - 32x^5 - 80x^4 + 32x^3 + 24x^2 - 6x - 1, ...
Triangle begins:
     1;
     2,   -1;
     4,   -2,    -1;
     8,   -4,    -4,    1;
    16,   -8,   -12,    4,    1;
    32,  -16,   -32,   12,    6,   -1;
    64,  -32,   -80,   32,   24,   -6,   -1;
   128,  -64,  -192,   80,   80,  -24,   -8,   1;
   256, -128,  -448,  192,  240,  -80,  -40,   8,   1;
   512, -256, -1024,  448,  672, -240, -160,  40,  10,  -1;
  1024, -512, -2304, 1024, 1792, -672, -560, 160,  60, -10,  -1;
  ...

J. C. Mason and D. C. Handscomb, Chebyshev polynomials, Chapman and Hall/CRC, 2002.

Paul Barry, On the Group of Almost-Riordan Arrays, arXiv preprint arXiv:1606.05077 [math.CO], 2016.

Cf. A180870, A028297. A008312, A155751.

A228565 := proc(n,k)
    local t,Vn,x ;
    t := arccos(x) ;
    Vn := cos((n+1/2)*t)/cos(t/2) ;
    coeftayl(%,x=0,n-k) ;
end proc:
for n from 0 to 10 do
    for k from 0 to n do
        printf("%d,",A228565(n,k)) ;
    end do:
    printf("\n") ;
end do: # R. J. Mathar, Mar 12 2014

V[n_] := Cos[(2*n + 1)*(ArcCos[x]/2)]/Cos[ArcCos[x]/2];
row[n_] := CoefficientList[V[n] + O[x]^(n + 1), x] // Reverse;
Table[row[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Nov 20 2017 *)

V(n+1,x) = 2xV(n,x) - V(n-1,x) with V(0,x) = 1, V(1,x) = 2x-1.
From Peter Bala, Jan 17 2014: (Start)
O.g.f. (1 - t)/(1 - 2*x*t + t^2) = 1 + (2*x - 1)*t +(4*x^2 - 2*x - 1)*t^2 + ....
In terms of the Chebyshev polynomials T(n,x) of the first kind and Chebyshev polynomials U(n,x) of the second kind we have
V(n,x) = U(n,x) - U(n-1,x);
V(n,x) + V(n-1,x) = 2*T(n,x);
V(n,x) = 1/u*T(2*n+1,u) with u = sqrt((1 + x)/2).
Also binomial(2*n,n)*V(n,x) = 2^(2*n)*Jacobi_P(n,-1/2,1/2,x). (End)

A053112 Expansion of (-1 + 1/(1-9x)^9)/(81x); related to A053108.

Original entry on oeis.org

1, 45, 1485, 40095, 938223, 19702683, 379980315, 6839645670, 116273976390, 1883638417518, 29282015399598, 439230230993970, 6385731819835410, 90312492880529370, 1246312401751305306, 16825217423642621631

Offset: 0

Wolfdieter Lang

G. C. Greubel, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (81, -2916, 61236, -826686, 7440174, -44641044, 172186884, -387420489, 387420489).

Cf. A053108, A053110 - A053113.

Without signs: A078812. With zeros: A049310. Cf. A008310 (T(n, x)), A008312 (U(n, x)).

[9^(n-1)*Binomial(n+9, 8): n in [0..30]]; // G. C. Greubel, Aug 16 2018

CoefficientList[Series[(-1+1/(1-9*x)^9)/(81*x),{x,0,30}],x] (* or *) LinearRecurrence[{81,-2916,61236,-826686,7440174,-44641044,172186884,-387420489,387420489}, {1,45,1485,40095,938223,19702683, 379980315, 6839645670,116273976390},20] (* Harvey P. Dale, Apr 27 2013 *)
Table[9^(n - 1)*Binomial[n + 9, 8], {n, 0, 30}] (* G. C. Greubel, Aug 16 2018 *)

vector(30,n,n--; 9^(n-1)*binomial(n+9, 8)) \\ G. C. Greubel, Aug 16 2018

G.f.: (-1 + 1/(1-9*x)^9)/(81*x).
a(n) = 9^(n-1)*binomial(n+9, 8).
a(0)=1, a(1)=45, a(2)=1485, a(3)=40095, a(4)=938223, a(5)=19702683, a(6)=379980315, a(7)=6839645670, a(8)=116273976390, a(n)=81*a(n-1)- 2916*a(n-2)+ 61236*a(n-3)- 826686*a(n-4)+ 7440174*a(n-5)- 44641044*a(n-6)+ 172186884*a(n-7)- 387420489*a(n-8)+ 387420489*a(n-9). - Harvey P. Dale, Apr 27 2013

A053118 Triangle of coefficients of Chebyshev's U(n,x) polynomials (exponents in decreasing order).

Original entry on oeis.org

1, 2, 0, 4, 0, -1, 8, 0, -4, 0, 16, 0, -12, 0, 1, 32, 0, -32, 0, 6, 0, 64, 0, -80, 0, 24, 0, -1, 128, 0, -192, 0, 80, 0, -8, 0, 256, 0, -448, 0, 240, 0, -40, 0, 1, 512, 0, -1024, 0, 672, 0, -160, 0, 10, 0, 1024, 0, -2304, 0, 1792, 0, -560, 0, 60, 0, -1, 2048, 0, -5120, 0, 4608, 0, -1792, 0, 280, 0, -12, 0, 4096, 0, -11264, 0, 11520, 0, -5376

Offset: 0

Wolfdieter Lang

a(n,m)= A053117(n,n-m) = 2^(n-m)*A049310(n,n-m).
G.f. for row polynomials U(n,x) (signed triangle): 1/(1-2*x*z+z^2). Unsigned triangle |a(n,m)| has Fibonacci polynomials F(n+1,2*x) as row polynomials with G.f. 1/(1-2*x*z-z^2).
Row sums (unsigned triangle) A000129(n+1) (Pell). Row sums (signed triangle) A000027(n+1) (natural numbers).

			1;
2,0;
4,0,-1;
8,0,-4,0;
16,0,-12,0,1;
... E.g. fourth row (n=3) {8,0,-4,0} corresponds to polynomial U(3,x)= 8*x^3-4*x.

Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.

Cf. A053117, A049310, A000129.

Triangle reflected without zeros: A008312 (the main entry).

Flatten[ Table[ Reverse[ CoefficientList[ ChebyshevU[n, x], x]], {n, 0, 12}]] (* Jean-François Alcover, Jan 20 2012 *)

a(n, m) := 0 if n= 0 is even else 0.

A123956 Triangle of coefficients of polynomials P(k) = 2XP(k - 1) - P(k - 2), P(0) = -1, P(1) = 1 + X, with twisted signs.

Original entry on oeis.org

-1, 1, 1, -1, -2, -2, 1, -3, 4, 4, -1, 4, 8, -8, -8, 1, 5, -12, -20, 16, 16, -1, -6, -18, 32, 48, -32, -32, 1, -7, 24, 56, -80, -112, 64, 64, -1, 8, 32, -80, -160, 192, 256, -128, -128, 1, 9, -40, -120, 240, 432, -448, -576, 256, 256, -1, -10, -50, 160, 400, -672, -1120, 1024, 1280, -512, -512

Offset: 0

Roger L. Bagula and Gary W. Adamson, Oct 27 2006

Up to signs also the coefficients of polynomials y(n+1) = y(n-1) - 2*h*y(n), arising when the ODE y' = -y is numerically solved with the leapfrog (a.k.a. two-step Nyström) method, with y(0) = 1, y(1) = 1 - h. In this case, the coefficients are negative exactly for the odd powers of h. - M. F. Hasler, Nov 30 2022

			Triangle begins:
  {-1},
  { 1,   1},
  {-1,  -2,  -2},
  { 1,  -3,   4,    4},
  {-1,   4,   8,   -8,   -8},
  { 1,   5, -12,  -20,   16,   16},
  {-1,  -6, -18,   32,   48,  -32,   -32},
  { 1,  -7,  24,   56,  -80, -112,    64,   64},
  {-1,   8,  32,  -80, -160,  192,   256, -128, -128},
  { 1,   9, -40, -120,  240,  432,  -448, -576,  256,  256},
  {-1, -10, -50,  160,  400, -672, -1120, 1024, 1280, -512, -512},
  ...

CRC Standard Mathematical Tables and Formulae, 16th ed. 1996, p. 484.
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 799.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Lutterbach, Approximating ODE y' = f(t,y) by using leapfrog method, Mathematics Stack Exchange, Nov 21 2019
Alastair MacDougall, 83.31 A Pascal-like triangle for coefficients of Chebyshev polynomials">, The Mathematical Gazette, Vol. 83, Issue 497 (Jul 1999), pp. 276-280.

Cf. A008312, A028297, A053117, A123235.

p[ -1, x] = 0; p[0, x] = 1; p[1, x] = x + 1;
p[k_, x_] := p[k, x] = 2*x*p[k - 1, x] - p[k - 2, x];
w = Table[CoefficientList[p[n, x], x], {n, 0, 10}];
An[d_] := Table[If[n == d && m  1/y)] /. 1/y -> 1, y], {d, 1, 11}] // Flatten

P=List([-1,1-'x]); {A123956(n,k)=for(i=#P, n+1, listput(P, P[i-1]-2*'x*P[i])); polcoef(P[n+1],k)*(-1)^((n-k-1)\2+!k*n\2)} \\ M. F. Hasler, Nov 30 2022

From M. F. Hasler, Nov 30 2022: (Start)
a(n,0) = (-1)^(n+1), a(n,1) = (-1)^floor(n/2)*n,
a(n,2) = (-1)^floor((n+1)/2)*A007590(n) = (-1)^floor((n+1)/2)*floor(n^2 / 2),
a(n,n) = a(n,n-1) = (-2)^(n-1) (n > 0),
a(n,3) / a(n,2) = { n/3 if n odd, -4*(n+2)/n if n even },
a(n,4) / a(n,3) = n/4 if n is even. (End)
From Peter Bala, Feb 06 2025: (Start)
Let T(n, x) and U(n, x) denote the n-th Chebyshev polynomial of the first and second kind. It appears that the row g.f.'s are as follows: for n >= 0,
row 4*n+1: T(4*n+1, x) + U(4*n, x); row 4*n+2: - 2 - T(4*n+2, x) - U(4*n+1, x);
row 4*n+3: 2 + T(4*n+3, x) + U(4*n+2, x); row 4*n+4: - T(4*n+4, x) - U(4*n+3, x). (End)

Offset changed to 0 by M. F. Hasler, Nov 30 2022

A075613 Triangular array read by rows, giving coefficients of P(n,X) = Product_{i=1..2n+1} (X - 1/cos(Pi*k/(2n+1))), a polynomial with integer coefficients.

Original entry on oeis.org

1, 1, -4, -4, 1, 1, -12, -12, 16, 16, 1, 1, -24, -24, 80, 80, -64, -64, 1, 1, -40, -40, 240, 240, -448, -448, 256, 256, 1, 1, -60, -60, 560, 560, -1792, -1792, 2304, 2304, -1024, -1024, 1, 1, -84, -84, 1120, 1120, -5376, -5376, 11520, 11520, -11264, -11264, 4096, 4096, 1, 1, -112, -112, 2016, 2016, -13440

Offset: 1

Benoit Cloitre, Oct 11 2002

Also, coefficients of U(2n,x) repeated (A008312, Chebyshev polynomials of second kind). - Ralf Stephan, Sep 06 2004

			P(3,X) = X^7 + X^6 - 24*X^5 - 24*X^4 + 80*X^3 + 80*X^2 - 64*X - 64.
Array begins:
  1, 1,  -4,  -4;
  1, 1, -12, -12, 16, 16;
  1, 1, -24, -24, 80, 80, -64, -64;
  ...

Cf. A008312, A075581.

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A123956 Triangle of coefficients of polynomials P(k) = 2XP(k - 1) - P(k - 2), P(0) = -1, P(1) = 1 + X, with twisted signs.