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 A130777 #64 Sep 19 2023 10:51:21
%S A130777 1,-1,1,-1,-1,1,1,-2,-1,1,1,2,-3,-1,1,-1,3,3,-4,-1,1,-1,-3,6,4,-5,-1,
%T A130777 1,1,-4,-6,10,5,-6,-1,1,1,4,-10,-10,15,6,-7,-1,1,-1,5,10,-20,-15,21,7,
%U A130777 -8,-1,1,-1,-5,15,20,-35,-21,28,8,-9,-1,1,1,-6,-15,35,35,-56,-28,36,9,-10,-1,1
%N A130777 Coefficients of first difference of Chebyshev S polynomials.
%C A130777 Inverse of triangle in A061554.
%C A130777 Signed version of A046854.
%C A130777 From _Paul Barry_, May 21 2009: (Start)
%C A130777 Riordan array ((1-x)/(1+x^2),x/(1+x^2)).
%C A130777 This triangle is the coefficient triangle for the Hankel transforms of the family of generalized Catalan numbers that satisfy a(n;r)=r*a(n-1;r)+sum{k=1..n-2, a(k)*a(n-1-k;r)}, a(0;r)=a(1;r)=1. The Hankel transform of a(n;r) is h(n)=sum{k=0..n, T(n,k)*r^k} with g.f. (1-x)/(1-r*x+x^2). These sequences include A086246, A000108, A002212. (End)
%C A130777 From _Wolfdieter Lang_, Jun 11 2011: (Start)
%C A130777 The Riordan array ((1+x)/(1+x^2),x/(1+x^2)) with entries Phat(n,k)= ((-1)^(n-k))*T(n,k) and o.g.f. Phat(x,z)=(1+z)/(1-x*z+z^2) for the row polynomials Phat(n,x) is related to Chebyshev C and S polynomials as follows.
%C A130777 Phat(n,x) = (R(n+1,x)-R(n,x))/(x+2) = S(2*n,sqrt(2+x))
%C A130777 with R(n,x)=C_n(x) in the Abramowitz and Stegun notation, p. 778, 22.5.11. See A049310 for the S polynomials. Proof from the o.g.f.s.
%C A130777 Recurrence for the row polynomials Phat(n,x):
%C A130777 Phat(n,x) = x*Phat(n-1,x) - Phat(n-2,x) for n>=1; Phat(-1,x)=-1, Phat(0,x)=1.
%C A130777 The A-sequence for this Riordan array Phat (see the W. Lang link under A006232 for A- and Z-sequences for Riordan matrices) is given by 1, 0, -1, 0, -1, 0, -2, 0, -5,.., starting with 1 and interlacing the negated A000108 with zeros (o.g.f. 1/c(x^2) = 1-c(x^2)*x^2, with the o.g.f. c(x) of A000108).
%C A130777 The Z-sequence has o.g.f. sqrt((1-2*x)/(1+2*x)), and it is given by A063886(n)*(-1)^n.
%C A130777 The A-sequence of the Riordan array T(n,k) is identical with the one for the Riordan array Phat, and the Z-sequence is -A063886(n).
%C A130777 (End)
%C A130777 The row polynomials P(n,x) are the characteristic polynomials of the adjacency matrices of the graphs which look like P_n (n vertices (nodes), n-1 lines (edges)), but vertex no. 1 has a loop. - _Wolfdieter Lang_, Nov 17 2011
%C A130777 From _Wolfdieter Lang_, Dec 14 2013: (Start)
%C A130777 The zeros of P(n,x) are x(n,j) = -2*cos(2*Pi*j/(2*n+1)), j=1..n. From P(n,x) = (-1)^n*S(2*n,sqrt(2-x)) (see, e.g., the Lemma 6 of the W. Lang link).
%C A130777 The discriminants of the P-polynomials are given in A052750. (End)
%D A130777 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964. Tenth printing, Wiley, 2002 (also electronically available).
%H A130777 Hyeong-Kwan Ju, <a href="https://doi.org/10.5831/HMJ.2017.39.4.665">On the sequence generated by a certain type of matrices</a>, Honam Math. J. 39, No. 4, 665-675 (2017), Theorem 2.16.
%H A130777 Wolfdieter Lang, <a href="http://arxiv.org/abs/1210.1018">The field Q(2cos(pi/n)), its Galois group and length ratios in the regular n-gon</a>, arXiv:1210.1018 [math.GR], 2012-2017; see Definition 1, Lemma 6 and Remark 4.
%H A130777 P. Steinbach, <a href="http://www.jstor.org/stable/2691048">Golden fields: a case for the heptagon</a>, Math. Mag. 70 (1997), p. 22-31 (formula 5).
%H A130777 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F A130777 Number triangle T(n,k) = (-1)^C(n-k+1,2)*C(floor((n+k)/2),k). - _Paul Barry_, May 21 2009
%F A130777 From _Wolfdieter Lang_, Jun 11 2011: (Start)
%F A130777 Row polynomials: P(n,x) = sum(k=0..n, T(n,k)*x^k) = R(2*n+1,sqrt(2+x)) / sqrt(2+x), with Chebyshev polynomials R with coefficients given in A127672 (scaled T-polynomials).
%F A130777 R(n,x) is called C_n(x) in Abramowitz and Stegun's handbook, p. 778, 22.5.11.
%F A130777 P(n,x) = S(n,x)-S(n-1,x), n>=0, S(-1,x)=0, with the Chebyshev S-polynomials (see the coefficient triangle A049310).
%F A130777 O.g.f. for row polynomials: P(x,z):= sum(n>=0, P(n,x)*z^n ) = (1-z)/(1-x*z+z^2).
%F A130777 (from the o.g.f. for R(2*n+1,x), n>=0, computed from the o.g.f. for the R-polynomials (2-x*z)/(1-x*z+z^2) (see A127672))
%F A130777 Proof of the Chebyshev connection from the o.g.f. for Riordan array property of this triangle (see the P. Barry comment above).
%F A130777 For the A- and Z-sequences of this Riordan array see a comment above. (End)
%F A130777 abs(T(n,k)) = A046854(n,k) = abs(A066170(n,k)) T(n,n-k) = A108299(n,k); abs(T(n,n-k)) = A065941(n,k). - _Johannes W. Meijer_, Aug 08 2011
%F A130777 From _Wolfdieter Lang_, Jul 31 2014: (Start)
%F A130777 Similar to the triangles A157751, A244419 and A180070 one can give for the row polynomials P(n,x) besides the usual three term recurrence another one needing only one recurrence step. This uses also a negative argument, namely P(n,x) = (-1)^(n-1)*(-1 + x/2)*P(n-1,-x) + (x/2)*P(n-1,x), n >= 1, P(0,x) = 1. Proof by computing the o.g.f. and comparing with the known one. This entails the alternative triangle recurrence T(n,k) = (-1)^(n-k)*T(n-1,k) + (1/2)*(1 + (-1)^(n-k))*T(n-1,k-1), n >= m >= 1, T(n,k) = 0 if n < k and T(n,0) = (-1)^floor((n+1)/2) = A057077(n+1). [P(n,x) recurrence corrected Aug 03 2014]
%F A130777 (End)
%e A130777 The triangle T(n,k) begins:
%e A130777 n\k 0 1 1 3 4 5 6 7 8 9 10 11 12 13 14 15 ...
%e A130777 0: 1
%e A130777 1: -1 1
%e A130777 2: -1 -1 1
%e A130777 3: 1 -2 -1 1
%e A130777 4: 1 2 -3 -1 1
%e A130777 5: -1 3 3 -4 -1 1
%e A130777 6: -1 -3 6 4 -5 -1 1
%e A130777 7: 1 -4 -6 10 5 -6 -1 1
%e A130777 8: 1 4 -10 -10 15 6 -7 -1 1
%e A130777 9: -1 5 10 -20 -15 21 7 -8 -1 1
%e A130777 10: -1 -5 15 20 -35 -21 28 8 -9 -1 1
%e A130777 11: 1 -6 -15 35 35 -56 -28 36 9 -10 -1 1
%e A130777 12: 1 6 -21 -35 70 56 -84 -36 45 10 -11 -1 1
%e A130777 13: -1 7 21 -56 -70 126 84 -120 -45 55 11 -12 -1 1
%e A130777 14: -1 -7 28 56 -126 -126 210 120 -165 -55 66 12 -13 -1 1
%e A130777 15: 1 -8 -28 84 126 -252 -210 330 165 -220 -66 78 13 -14 -1 1
%e A130777 ... reformatted and extended - _Wolfdieter Lang_, Jul 31 2014.
%e A130777 ---------------------------------------------------------------------------
%e A130777 From _Paul Barry_, May 21 2009: (Start)
%e A130777 Production matrix is
%e A130777 -1, 1,
%e A130777 -2, 0, 1,
%e A130777 -2, -1, 0, 1,
%e A130777 -4, 0, -1, 0, 1,
%e A130777 -6, -1, 0, -1, 0, 1,
%e A130777 -12, 0, -1, 0, -1, 0, 1,
%e A130777 -20, -2, 0, -1, 0, -1, 0, 1,
%e A130777 -40, 0, -2, 0, -1, 0, -1, 0, 1,
%e A130777 -70, -5, 0, -2, 0, -1, 0, -1, 0, 1 (End)
%e A130777 Row polynomials as first difference of S polynomials:
%e A130777 P(3,x) = S(3,x) - S(2,x) = (x^3 - 2*x) - (x^2 -1) = 1 - 2*x - x^2 +x^3.
%e A130777 Alternative triangle recurrence (see a comment above): T(6,2) = T(5,2) + T(5,1) = 3 + 3 = 6. T(6,3) = -T(5,3) + 0*T(5,1) = -(-4) = 4. - _Wolfdieter Lang_, Jul 31 2014
%p A130777 A130777 := proc(n,k): (-1)^binomial(n-k+1,2)*binomial(floor((n+k)/2),k) end: seq(seq(A130777(n,k), k=0..n), n=0..11); # _Johannes W. Meijer_, Aug 08 2011
%t A130777 T[n_, k_] := (-1)^Binomial[n - k + 1, 2]*Binomial[Floor[(n + k)/2], k];
%t A130777 Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Nov 14 2017, from Maple *)
%o A130777 (Sage)
%o A130777 @CachedFunction
%o A130777 def A130777(n,k):
%o A130777 if n< 0: return 0
%o A130777 if n==0: return 1 if k == 0 else 0
%o A130777 h = A130777(n-1,k) if n==1 else 0
%o A130777 return A130777(n-1,k-1) - A130777(n-2,k) - h
%o A130777 for n in (0..9): [A130777(n,k) for k in (0..n)] # _Peter Luschny_, Nov 20 2012
%Y A130777 Cf. A066170, A046854, A057077 (first column).
%Y A130777 Row sums: A010892(n+1); repeat(1,0,-1,-1,0,1). Alternating row sums: A061347(n+2); repeat(1,-2,1).
%K A130777 sign,tabl,easy
%O A130777 0,8
%A A130777 _Philippe Deléham_, Jul 14 2007
%E A130777 New name and Chebyshev comments by _Wolfdieter Lang_, Jun 11 2010