This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A039991 #70 Mar 16 2025 10:09:33
%S A039991 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,
%T A039991 64,0,-112,0,56,0,-7,0,128,0,-256,0,160,0,-32,0,1,256,0,-576,0,432,0,
%U A039991 -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
%N A039991 Triangle of coefficients of cos(x)^n in polynomial for cos(nx).
%C A039991 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.
%C A039991 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
%D A039991 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.
%D A039991 Martin Aigner and Gunter M. Ziegler, Proofs From the Book, Springer 2004. See Chapter 18, Appendix.
%D A039991 E. A. Guilleman, Synthesis of Passive Networks, Wiley, 1957, p. 593.
%D A039991 Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.
%H A039991 T. D. Noe, <a href="/A039991/b039991.txt">Table of n, a(n) for n = 0..5150</a>
%H A039991 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A039991 Daniel J. Greenhoe, <a href="https://www.researchgate.net/publication/337858762_Frames_and_Bases_Structure_and_Design_version_020">Frames and Bases: Structure and Design</a>, Version 0.20, Signal Processing ABCs series (2019) Vol. 4, see page 172.
%H A039991 Daniel J. Greenhoe, <a href="https://www.researchgate.net/publication/337858659_A_Book_Concerning_Transforms_version_010">A Book Concerning Transforms</a>, Version 0.10, Signal Processing ABCs series (2019) Vol. 5, see page 94.
%H A039991 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F A039991 T(n, m) = 0 if n<m or m odd, (-1)^(m/2) if m=n is even, ((-1)^(3*m/2))*(2^(n-m-1))*n*binomial(n-1-m/2, n-1-m)/(n-m) else T(n, m) = 2*T(n-1, m) - T(n-2, m-2), n >= 2, m >= 0; T(n, -2) = T(n, -1) = 0, T(0, 0) = T(1, 0) = 1.
%F A039991 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.
%F A039991 G.f. row polynomials: (1-z)/(1 - 2*z + (x*z)^2). - _Wolfdieter Lang_, Aug 06 2014
%F A039991 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
%F A039991 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
%e A039991 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.
%e A039991 From _Wolfdieter Lang_, Aug 06 2014: (Start)
%e A039991 The triangle a(n,m) begins:
%e A039991 n\m 0 1 2 3 4 5 6 7 8 9 10 11 ...
%e A039991 0: 1
%e A039991 1: 1 0
%e A039991 2: 2 0 -1
%e A039991 3: 4 0 -3 0
%e A039991 4: 8 0 -8 0 1
%e A039991 5: 16 0 -20 0 5 0
%e A039991 6: 32 0 -48 0 18 0 -1
%e A039991 7: 64 0 -112 0 56 0 -7 0
%e A039991 8: 128 0 -256 0 160 0 -32 0 1
%e A039991 9: 256 0 -576 0 432 0 -120 0 9 0
%e A039991 10: 512 0 -1280 0 1120 0 -400 0 50 0 -1
%e A039991 11: 1024 0 -2816 0 2816 0 -1232 0 220 0 -11 0
%e A039991 ...
%e A039991 --------------------------------------------------------------------------
%e A039991 Chebyshev T-polynomials (decreasing even or odd powers):
%e A039991 n=3: T(3, x) = 4*x^3 - 3*x^1; n=4: T(4, x) = 8*x^4 - 8*x^2 + 1. (End)
%p A039991 seq(seq(coeff(orthopoly[T](i,x),x,i-j),j=0..i),i=0..20); # _Robert Israel_, Aug 07 2014
%t A039991 row[n_] := CoefficientList[ ChebyshevT[n, x], x] // Reverse; Table[row[n], {n, 0, 11}] // Flatten(* _Jean-François Alcover_, Sep 14 2012 *)
%o A039991 (Magma)
%o A039991 function T(n,k) // T = A039991
%o A039991 if k lt 0 or k gt n then return 0;
%o A039991 elif n lt 2 and k eq 0 then return 1;
%o A039991 else return 2*T(n-1, k) - T(n-2, k-2);
%o A039991 end if; return T;
%o A039991 end function;
%o A039991 [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Aug 10 2022
%o A039991 (SageMath)
%o A039991 def T(n, k): # T = A039991
%o A039991 if (n<2 and k==0): return 1
%o A039991 elif (k<0 or k>n): return 0
%o A039991 else: return 2*T(n-1, k) - T(n-2, k-2)
%o A039991 flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Aug 10 2022
%o A039991 (PARI) 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 */
%Y A039991 Cf. A028297 (without vanishing columns). A008310 (zero columns deleted then rows reversed).
%Y A039991 Triangle without zeros: A028297. Without signs: A081265.
%Y A039991 Cf. A053120 (increasing powers of x).
%K A039991 tabl,easy,sign,nice
%O A039991 0,4
%A A039991 _David W. Wilson_
%E A039991 Entry improved by comments from _Wolfdieter Lang_, Jan 11 2000.
%E A039991 Edited: A053120 added in comment and crossrefs. Cfs. A028297 and A008310 specified. - _Wolfdieter Lang_, Aug 06 2014