A081265 -id:A081265 - cp's OEIS frontend

A039991 Triangle of coefficients of cos(x)^n in polynomial for cos(nx).

1, 1, 0, 2, 0, -1, 4, 0, -3, 0, 8, 0, -8, 0, 1, 16, 0, -20, 0, 5, 0, 32, 0, -48, 0, 18, 0, -1, 64, 0, -112, 0, 56, 0, -7, 0, 128, 0, -256, 0, 160, 0, -32, 0, 1, 256, 0, -576, 0, 432, 0, -120, 0, 9, 0, 512, 0, -1280, 0, 1120, 0, -400, 0, 50, 0, -1, 1024, 0, -2816, 0, 2816, 0, -1232, 0, 220, 0, -11, 0

Offset: 0

David W. Wilson

Also triangle of coefficients of Chebyshev polynomials of first kind (T(n,x)) in decreasing order of powers of x. A053120 gives the coefficients in increasing order.

The polynomials R(n,x) := Sum_{m=0..n} a(n,m)*sqrt(x)^m, have g.f. (1-z)/(1 - 2*z + x*z^2) = ((1-z)/(1-2*z))/(1 - x*(-z^2/(1-2*z))) (from the row reversion of the g.f. of A053120 and x^2 -> x). Therefore this triangle becomes the Riordan triangle ((1-z)/(1-2*z), -z^2/(1-2*z)) if the vanishing columns are deleted (see A028297) and zeros are appended in each row numbered n>=1 in order to obtain a triangle. This is then A201701 with negative odd numbered columns. - Wolfdieter Lang, Aug 06 2014

			Letting c = cos x, we have: cos 0x = 1, cos 1x = 1c; cos 2x = 2c^2-1; cos 3x = 4c^3-3c, cos 4x = 8c^4-8c^2+1, etc.
From _Wolfdieter Lang_, Aug 06 2014: (Start)
The triangle a(n,m) begins:
  n\m    0 1     2 3    4 5     6 7   8 9  10 11 ...
  0:     1
  1:     1 0
  2:     2 0    -1
  3:     4 0    -3 0
  4:     8 0    -8 0    1
  5:    16 0   -20 0    5 0
  6:    32 0   -48 0   18 0    -1
  7:    64 0  -112 0   56 0    -7 0
  8:   128 0  -256 0  160 0   -32 0   1
  9:   256 0  -576 0  432 0  -120 0   9 0
  10:  512 0 -1280 0 1120 0  -400 0  50 0  -1
  11: 1024 0 -2816 0 2816 0 -1232 0 220 0 -11  0
  ...
--------------------------------------------------------------------------
Chebyshev T-polynomials (decreasing even or odd powers):
n=3: T(3, x) = 4*x^3 - 3*x^1; n=4: T(4, x) = 8*x^4 - 8*x^2 + 1. (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.
Martin Aigner and Gunter M. Ziegler, Proofs From the Book, Springer 2004. See Chapter 18, Appendix.
E. A. Guilleman, Synthesis of Passive Networks, Wiley, 1957, p. 593.
Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.

T. D. Noe, Table of n, a(n) for n = 0..5150
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].
Daniel J. Greenhoe, Frames and Bases: Structure and Design, Version 0.20, Signal Processing ABCs series (2019) Vol. 4, see page 172.
Daniel J. Greenhoe, A Book Concerning Transforms, Version 0.10, Signal Processing ABCs series (2019) Vol. 5, see page 94.
Index entries for sequences related to Chebyshev polynomials.

Cf. A028297 (without vanishing columns). A008310 (zero columns deleted then rows reversed).
Triangle without zeros: A028297. Without signs: A081265.
Cf. A053120 (increasing powers of x).

function T(n,k) // T = A039991
  if k lt 0 or k gt n then return 0;
  elif n lt 2 and k eq 0 then return 1;
  else return 2*T(n-1, k) - T(n-2, k-2);
  end if; return T;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 10 2022

seq(seq(coeff(orthopoly[T](i,x),x,i-j),j=0..i),i=0..20); # Robert Israel, Aug 07 2014

row[n_] := CoefficientList[ ChebyshevT[n, x], x] // Reverse; Table[row[n], {n, 0, 11}] // Flatten(* Jean-François Alcover, Sep 14 2012 *)

T(n,m)=(1+(-1)^m)*(binomial(n-m/2,n-m)+binomial(n-1-m/2,n-m))*2^(n-m-2)*(-1)^((m+1-(-1)^m)/2) /* Tani Akinari, Jul 18 2024 */

def T(n, k): # T = A039991
    if (n<2 and k==0): return 1
    elif (k<0 or k>n): return 0
    else: return 2*T(n-1, k) - T(n-2, k-2)
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Aug 10 2022

T(n, m) = 0 if n= 2, m >= 0; T(n, -2) = T(n, -1) = 0, T(0, 0) = T(1, 0) = 1.
G.f. for m-th column: 0 if m odd, (1-x)/(1-2*x) if m=0, else ((-1)^(m/2))*(x^m)*(1-x)/(1-2*x)^(m/2+1). For g.f. for row polynomials and row sums, see A053120.
G.f. row polynomials: (1-z)/(1 - 2*z + (x*z)^2). - Wolfdieter Lang, Aug 06 2014
Recurrence for the row polynomials Trev(n, x):= x^n*T(n, 1/x) = Sum_{m=0..n} T(n, m)*x^m; Trev(n, x) = 2*Trev(n-1, x) - x^2*Trev(n-2, x), n >= 1, Trev(-1, x) = 1/x^2 and Trev(0, x) = 1. From the T(n, x) recurrence. Compare this with A081265. - Wolfdieter Lang, Aug 07 2014
T(n,m) = (1+(-1)^m)*(binomial(n-m/2,n-m)+binomial(n-1-m/2,n-m))*2^(n-m-2)*(-1)^((m+1-(-1)^m)/2). - Tani Akinari, Jul 18 2024

Entry improved by comments from Wolfdieter Lang, Jan 11 2000.

Edited: A053120 added in comment and crossrefs. Cfs. A028297 and A008310 specified. - Wolfdieter Lang, Aug 06 2014

A201701 Riordan triangle ((1-x)/(1-2x), x^2/(1-2x)).

Original entry on oeis.org

1, 1, 0, 2, 1, 0, 4, 3, 0, 0, 8, 8, 1, 0, 0, 16, 20, 5, 0, 0, 0, 32, 48, 18, 1, 0, 0, 0, 64, 112, 56, 7, 0, 0, 0, 0, 128, 256, 160, 32, 1, 0, 0, 0, 0, 256, 576, 432, 120, 9, 0, 0, 0, 0, 0, 512, 1280, 1120, 400, 50, 1, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Dec 03 2011

Triangle T(n,k), read by rows, given by (1,1,0,0,0,0,0,0,0,...) DELTA (0,1,-1,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.
Skewed version of triangle in A200139.
Triangle without zeros: A207537.
For the version with negative odd numbered columns, which is Riordan ((1-x)/(1-2*x), -x^2/(1-2*x)) see comments on A028297 and A039991. - Wolfdieter Lang, Aug 06 2014
This is an example of a stretched Riordan array in the terminology of Section 2 of Corsani et al. - Peter Bala, Jul 14 2015

			The triangle T(n,k) begins:
  n\k      0     1     2     3     4    5   6  7 8 9 10 11 ...
  0:       1
  1:       1     0
  2:       2     1     0
  3:       4     3     0     0
  4:       8     8     1     0     0
  5:      16    20     5     0     0    0
  6:      32    48    18     1     0    0   0
  7:      64   112    56     7     0    0   0  0
  8:     128   256   160    32     1    0   0  0 0
  9:     256   576   432   120     9    0   0  0 0 0
  10:    512  1280  1120   400    50    1   0  0 0 0  0
  11:   1024  2816  2816  1232   220   11   0  0 0 0  0  0
  ...  reformatted and extended. - _Wolfdieter Lang_, Aug 06 2014

C. Corsani, D. Merlini, and R. Sprugnoli, Left-inversion of combinatorial sums, Discrete Mathematics, 180 (1998) 107-122.

Columns include A011782, A001792, A001793, A001794, A006974, A006975, A006976.
Diagonals sums are in A052980.
Cf. A028297, A081265, A124182, A131577, A039991 (zero-columns deleted, unsigned and zeros appended).
Cf. A098158, A200139, A207537.
Cf. A028297 (signed version, zeros deleted). Cf. A034839.

(* The function RiordanArray is defined in A256893. *)
RiordanArray[(1 - #)/(1 - 2 #)&, #^2/(1 - 2 #)&, 11] // Flatten (* Jean-François Alcover, Jul 16 2019 *)

T(n,k) = 2*T(n-1,k) + T(n-2,k-1) with T(0,0) = T(1,0) = 1, T(1,1) = 0 and T(n,k) = 0 for k<0 or for n

Sum_{k=0..n} T(n,k)^2 = A002002(n) for n>0.

Sum_{k=0..n} T(n,k)*x^k = A138229(n), A006495(n), A138230(n), A087455(n), A146559(n), A000012(n), A011782(n), A001333(n), A026150(n), A046717(n), A084057(n), A002533(n), A083098(n), A084058(n), A003665(n), A002535(n), A133294(n), A090042(n), A125816(n), A133343(n), A133345(n), A120612(n), A133356(n), A125818(n) for x = -6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17 respectively.

G.f.: (1-x)/(1-2*x-y*x^2). - Philippe Deléham, Mar 03 2012

From Peter Bala, Jul 14 2015: (Start)

Factorizes as A034839 * A007318 = (1/(1 - x), x^2/(1 - x)^2) * (1/(1 - x), x/(1 - x)) as a product of Riordan arrays.

T(n,k) = Sum_{i = k..floor(n/2)} binomial(n,2*i) *binomial(i,k). (End)

Name changed, keyword:easy added, crossrefs A028297 and A039991 added, and g.f. corrected by Wolfdieter Lang, Aug 06 2014

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A039991 Triangle of coefficients of cos(x)^n in polynomial for cos(nx).

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A201701 Riordan triangle ((1-x)/(1-2x), x^2/(1-2x)).

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cp's OEIS Frontend

A039991 Triangle of coefficients of cos(x)^n in polynomial for cos(nx).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

SageMath

Formula

Extensions

A201701 Riordan triangle ((1-x)/(1-2*x), x^2/(1-2*x)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A201701 Riordan triangle ((1-x)/(1-2x), x^2/(1-2x)).