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A132460 Irregular triangle read by rows of the initial floor(n/2) + 1 coefficients of 1/C(x)^n, where C(x) is the g.f. of the Catalan sequence (A000108).

%I A132460 #68 Jan 05 2025 19:51:38
%S A132460 1,1,1,-2,1,-3,1,-4,2,1,-5,5,1,-6,9,-2,1,-7,14,-7,1,-8,20,-16,2,1,-9,
%T A132460 27,-30,9,1,-10,35,-50,25,-2,1,-11,44,-77,55,-11,1,-12,54,-112,105,
%U A132460 -36,2,1,-13,65,-156,182,-91,13,1,-14,77,-210,294,-196,49,-2
%N A132460 Irregular triangle read by rows of the initial floor(n/2) + 1 coefficients of 1/C(x)^n, where C(x) is the g.f. of the Catalan sequence (A000108).
%C A132460 The length of row n is A008619(n).
%C A132460 Essentially equals a signed version of A034807, the triangle of Lucas polynomials. The initial n coefficients of 1/C(x)^n consist of row n followed by floor((n-1)/2) zeros for n > 0.
%C A132460 For the following formula for 1/C(x)^n see the W. Lang reference, proposition 1 on p. 411:
%C A132460 1/C(x)^n = (sqrt(x))^n*(S(n,1/sqrt(x)) - sqrt(x)*S(n-1,1/sqrt(x))*C(x)), n >= 0, with the Chebyshev polynomials S(n,x) with coefficients given in A049310. See also the coefficient array A115139 for P(n,x) = (sqrt(x)^(n-1))*S(n-1, 1/sqrt(x)). - _Wolfdieter Lang_, Sep 14 2013
%C A132460 This triangular array is composed of interleaved rows of reversed, A127677 (cf. A156308, A217476, A263916) and reversed, signed A111125. - _Tom Copeland_, Nov 07 2015
%C A132460 It seems that the n-th row lists the coefficients of the HOMFLYPT (HOMFLY) polynomial reduced to one variable for link family n, see Jablan's slide 38. - _Andrey Zabolotskiy_, Jan 16 2018
%C A132460 For n >= 1 row n gives the coefficients of the Girard-Waring formula for the sum of x1^n + x2^n in terms of the elementary symmetric functions e_1(x1,x2) = x1 + x2 and e_2(x1,x2) = x1*x2. This is an array using the partitions of n, in the reverse Abramowitz-Stegun order, with all partitions with parts larger than 2 eliminated. E.g., n = 4: x1^4 + x2^4 = 1*e1^4 - 4*e1^3*e2 + 2*e1*e2^2. See also A115131, row n = 4, with the mentioned partitions omitted. - _Wolfdieter Lang_, May 03 2019
%C A132460 Row n lists the coefficients of the n-th Faber polynomial for the replicable function given in A154272 with offset -1. - _Ben Toomey_, May 12 2020
%H A132460 Michael De Vlieger, <a href="/A132460/b132460.txt">Table of n, a(n) for n = 0..10200</a> (rows 0 <= n <= 200, flattened)
%H A132460 Tom Copeland, <a href="https://tcjpn.wordpress.com/2015/10/12/the-elliptic-lie-triad-kdv-and-ricattt-equations-infinigens-and-elliptic-genera/">Addendum to Elliptic Lie Triad</a>
%H A132460 G. Dattoli, E. Di Palma and E. Sabia, <a href="http://dx.doi.org/10.1007/s00006-014-0463-3">Cardan Polynomials, Chebyshev Exponents, Ultra-Radicals and Generalized Imaginary Units</a>, Advances in Applied Clifford Algebras, 2014.
%H A132460 Pentti Haukkanen, Jorma Merikoski and Seppo Mustonen, <a href="http://www.acta.sapientia.ro/acta-math/C6-2/math62-5.pdf">Some polynomials associated with regular polygons</a>, Acta Univ. Sapientiae, Mathematica, 6, 2 (2014) 178-193.
%H A132460 S. Jablan, <a href="http://indico.ictp.it/event/a08157/session/35/contribution/19/material/0/0.pdf">Knots, computers, conjectures</a>
%H A132460 Wolfdieter Lang, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/38-5/lang.pdf">On polynomials related to powers of the generating function of Catalan's numbers</a>, Fib. Quart. 38 (2000) 408-419. Eq. (23) with n -> -n and eq. (20).
%H A132460 Jorma K. Merikoski, <a href="https://doi.org/10.7546/nntdm.2021.27.2.79-87">Regular polygons, Morgan-Voyce polynomials, and Chebyshev polynomials</a>, Notes on Num. Theor. and Disc. Math. (2021) Vol. 27, No. 2, 79-87.
%F A132460 T(n,k) = (-1)^k*( C(n-k,k) + C(n-k-1,k-1) ) for n >= 0, 0 <= k <= floor(n/2).
%F A132460 T(0,0) = 1; T(n,k) = (-1)^k*n*binomial(n-k,k)/(n-k), k = 0..floor(n/2). - _Wolfdieter Lang_, May 03 2019
%e A132460 The irregular triangle T(n,k) begins:
%e A132460 n\k 0    1    2    3    4    5    6   7 ...
%e A132460 -------------------------------------------------
%e A132460 0:  1
%e A132460 1:  1
%e A132460 2:  1   -2
%e A132460 3:  1   -3
%e A132460 4:  1   -4    2
%e A132460 5:  1   -5    5
%e A132460 6:  1   -6    9   -2
%e A132460 7:  1   -7   14   -7
%e A132460 8:  1   -8   20  -16    2
%e A132460 9:  1   -9   27  -30    9
%e A132460 10: 1  -10   35  -50   25   -2
%e A132460 11: 1  -11   44  -77   55  -11
%e A132460 12: 1  -12   54 -112  105  -36    2
%e A132460 13: 1  -13   65 -156  182  -91   13
%e A132460 14: 1  -14   77 -210  294 -196   49  -2
%e A132460 ... (reformatted - _Wolfdieter Lang_, May 03 2019)
%t A132460 T[0, 0] = 1; T[n_, k_] := (-1)^k (Binomial[n-k, k] + Binomial[n-k-1, k-1]);
%t A132460 Table[T[n, k], {n, 0, 14}, {k, 0, n/2}] // Flatten (* _Jean-François Alcover_, Jun 04 2018 *)
%o A132460 (PARI) {T(n,k)=if(k>n\2,0,(-1)^k*(binomial(n-k, k)+binomial(n-k-1, k-1)))}
%Y A132460 Cf. A000108, A008619, A034807 (Lucas polynomials), A111125, A115131 (Waring numbers), A127677, A132461 (row squared sums), A156308, A217476, A263916.
%K A132460 sign,easy,tabf
%O A132460 0,4
%A A132460 _Paul D. Hanna_, Aug 21 2007