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 A180870 #76 Oct 07 2024 09:03:36
%S A180870 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,
%T A180870 24,6,-1,128,64,-192,-80,80,24,-8,-1,256,128,-448,-192,240,80,-40,-8,
%U A180870 1,512,256,-1024,-448,672,240,-160,-40,10,1
%N 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)).
%C A180870 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)).
%C A180870 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
%D A180870 J. C. Mason and D. C. Handscomb, Chebyshev polynomials, Chapman and Hall/CRC 2002.
%H A180870 Paul Barry, <a href="https://arxiv.org/abs/1606.05077">On the Group of Almost-Riordan Arrays</a>, arXiv preprint arXiv:1606.05077 [math.CO], 2016.
%H A180870 Wikipedia, <a href="http://en.wikipedia.org/wiki/Dirichlet_kernel">Dirichlet kernel</a>.
%F A180870 From _Peter Bala_, Jan 17 2014: (Start)
%F A180870 O.g.f. (1 + t)/(1 - 2*x*t + t^2) = 1 + (2*x + 1)*t + (4*x^2 + 2*x - 1)*t^2 + ....
%F A180870 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.
%F A180870 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)
%F A180870 Row sums: 2*n+1. - _Michel Marcus_, Jul 16 2014
%F A180870 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
%F A180870 From _Wolfdieter Lang_, Jul 30 2014: (Start)
%F A180870 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).
%F A180870 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.
%F A180870 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)
%e A180870 The triangle T(n,m) begins:
%e A180870 n\m 0 1 2 3 4 5 6 7 8 9 10 ...
%e A180870 0: 1
%e A180870 1: 2 1
%e A180870 2: 4 2 -1
%e A180870 3: 8 4 -4 -1
%e A180870 4: 16 8 -12 -4 1
%e A180870 5: 32 16 -32 -12 6 1
%e A180870 6: 64 32 -80 -32 24 6 -1
%e A180870 7: 128 64 -192 -80 80 24 -8 -1
%e A180870 8: 256 128 -448 -192 240 80 -40 -8 1
%e A180870 9: 512 256 -1024 -448 672 240 -160 -40 10 1
%e A180870 10: 1024 512 -2304 -1024 1792 672 -560 -160 60 10 -1
%e A180870 ... reformatted - _Wolfdieter Lang_, Jul 26 2014
%e A180870 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
%p A180870 ogf := (1 + t)/(1 - 2*x*t + t^2):
%p A180870 ser := simplify(series(ogf, t, 12)): tc := n -> coeff(ser, t, n):
%p A180870 Trow := n -> local k; seq(coeff(tc(n), x, n-k), k = 0..n):
%p A180870 seq(print(Trow(n)), n = 0..9); # _Peter Luschny_, Oct 07 2024
%o A180870 (PARI) 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
%Y A180870 Cf. A008312, A028297, A157751, A228565, A049310, A244419 (row reversed triangle).
%K A180870 easy,sign,tabl
%O A180870 0,2
%A A180870 _Jonny Griffiths_, Sep 21 2010
%E A180870 Missing term in sequence corrected by _Paul Curtz_, Dec 31 2011
%E A180870 Edited (name reformulated, Wikipedia link added) by _Wolfdieter Lang_, Jul 26 2014