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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).
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%I #66 Mar 21 2024 20:57:31

%S 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 27,-30,9,1,-10,35,-50,25,-2,1,-11,44,-77,55,-11,1,-12,54,-112,105,

%U -36,2,1,-13,65,-156,182,-91,13,1,-14,77,-210,294,-196,49,-2

%N 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 The length of row n is A008619(n).

%C 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 For the following formula for 1/C(x)^n see the W. Lang reference, proposition 1 on p. 411:

%C 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 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 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 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 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 Michael De Vlieger, <a href="/A132460/b132460.txt">Table of n, a(n) for n = 0..10200</a> (rows 0 <= n <= 200, flattened)

%H 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 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 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 S. Jablan, <a href="http://indico.ictp.it/event/a08157/session/35/contribution/19/material/0/0.pdf">Knots, computers, conjectures</a>

%H Wolfdieter Lang, <a href="http://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 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 T(n,k) = (-1)^k*( C(n-k,k) + C(n-k-1,k-1) ) for n >= 0, 0 <= k <= floor(n/2).

%F 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 The irregular triangle T(n,k) begins:

%e n\k 0 1 2 3 4 5 6 7 ...

%e -------------------------------------------------

%e 0: 1

%e 1: 1

%e 2: 1 -2

%e 3: 1 -3

%e 4: 1 -4 2

%e 5: 1 -5 5

%e 6: 1 -6 9 -2

%e 7: 1 -7 14 -7

%e 8: 1 -8 20 -16 2

%e 9: 1 -9 27 -30 9

%e 10: 1 -10 35 -50 25 -2

%e 11: 1 -11 44 -77 55 -11

%e 12: 1 -12 54 -112 105 -36 2

%e 13: 1 -13 65 -156 182 -91 13

%e 14: 1 -14 77 -210 294 -196 49 -2

%e ... (reformatted - _Wolfdieter Lang_, May 03 2019)

%t T[0, 0] = 1; T[n_, k_] := (-1)^k (Binomial[n-k, k] + Binomial[n-k-1, k-1]);

%t Table[T[n, k], {n, 0, 14}, {k, 0, n/2}] // Flatten (* _Jean-François Alcover_, Jun 04 2018 *)

%o (PARI) {T(n,k)=if(k>n\2,0,(-1)^k*(binomial(n-k, k)+binomial(n-k-1, k-1)))}

%Y Cf. A000108, A008619, A034807 (Lucas polynomials), A111125, A115131 (Waring numbers), A127677, A132461 (row squared sums), A156308, A217476, A263916.

%K sign,easy,tabf

%O 0,4

%A _Paul D. Hanna_, Aug 21 2007