A127677 -id:A127677 - cp's OEIS frontend

1, 1, 2, 1, 4, 2, 1, 6, 9, 2, 1, 8, 20, 16, 2, 1, 10, 35, 50, 25, 2, 1, 12, 54, 112, 105, 36, 2, 1, 14, 77, 210, 294, 196, 49, 2, 1, 16, 104, 352, 660, 672, 336, 64, 2, 1, 18, 135, 546, 1287, 1782, 1386, 540, 81, 2, 1, 20, 170, 800, 2275, 4004, 4290, 2640, 825, 100, 2

Offset: 0

Gary W. Adamson, May 29 2003

Sum of row #n = A000204(2n). (But sum of row #0 = 1.)
Row #n has the unsigned coefficients of the monic polynomial whose roots are 2 cos(Pi*(2k-1)/(4n)) for k=1..2n. [Comment corrected by Barry Brent, Jan 03 2006]
The positive roots are some diagonal lengths of a regular (4n)-gon, inscribed in the unit circle.
Polynomial of row #n = Sum_{m=0..n} (-1)^m * T(n,m) x^(2*n-2*m).
This is the unsigned version of the coefficient table for scaled Chebyshev T(2*n,x) polynomials. - Wolfdieter Lang, Mar 07 2007
Reversed A127677 (cf. A156308, A217476, A263916). - Tom Copeland, Nov 07 2015

			First few Chebyshev T(2*n,x) polynomials:
  T(2*0,x) = 1;
  T(2*1,x) = x^2 -   2;
  T(2*2,x) = x^4 -   4*x^2 +  2;
  T(2*3,x) = x^6 -   6*x^4 +  9*x^2 -  2;
  T(2*4,x) = x^8 -   8*x^6 + 20*x^4 - 16*x^2 +  2;
  T(2*5,x) = x^10 - 10*x^8 + 35*x^6 - 50*x^4 + 25*x^2 - 2;
Triangle begins as:
  1;
  1,  2;
  1,  4,  2;
  1,  6,  9,   2;
  1,  8, 20,  16,   2;
  1, 10, 35,  50,  25,  2;
  1, 12, 54, 112, 105, 36, 2;

I. Kaplansky and J. Riordan, The problème des ménages, Scripta Math. 12, (1946), 113-124. See p. 118.
Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990. p. 37, eq.(1.96) and p. 4. eq.(1.10).

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
I. Kaplansky and J. Riordan, The problème des ménages, Scripta Math. 12, (1946), 113-124. [Scan of annotated copy]

Row sums are A005248 for n > 0.
Companion triangle A082985.
Cf. A082985 (unsigned scaled coefficient table for Chebyshev's T(2*n+1, x) polynomials).
Cf. A127677, A156308, A217476, A263916.

A084534:= func< n,k | k eq 0 select 1 else 2*(n/k)*Binomial(2*n-k-1, k-1) >;
[A084534(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 02 2022

T := proc(n, m): if n=0 then 1 else binomial(2*n-m, m)*2*n/(2*n-m) fi: end: seq(seq(T(n,m),m=0..n),n=0..10); # Johannes W. Meijer, May 31 2018

a[n_, m_] := Binomial[2n-m, m]*2n/(2n-m); a[0, 0] = 1; Table[a[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Jean-François Alcover, Apr 12 2016, after Wolfdieter Lang *)

T(n,m) = if(n==0, m==0, binomial(2*n-m, m)*2*n/(2*n-m)) \\ Andrew Howroyd, Dec 18 2017

def A084534(n,k): return 1 if (k==0) else 2*(n/k)*binomial(2*n-k-1, k-1)
flatten([[A084534(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 02 2022

T(n,m) = binomial(2*n-m, m)*2*n/(2*n-m) for n > 0. - Andrew Howroyd, Dec 18 2017
Signed version from Wolfdieter Lang, Mar 07 2007: (Start)
a(n,m)=0 if na(n,m)=0 if na(n,m)=0 if nA127674(n,n-m)/2^(2*(n-m)-1) (scaled coefficients of Chebyshev's T(2*n,x), decreasing even powers). [corrected by Johannes W. Meijer, May 31 2018] (End)

Edited by Don Reble, Nov 12 2005

A219233 Alternating row sums of Riordan triangle A110162.

Original entry on oeis.org

1, -3, 7, -18, 47, -123, 322, -843, 2207, -5778, 15127, -39603, 103682, -271443, 710647, -1860498, 4870847, -12752043, 33385282, -87403803, 228826127, -599074578, 1568397607, -4106118243, 10749957122, -28143753123, 73681302247, -192900153618, 505019158607

Offset: 0

Wolfdieter Lang, Nov 16 2012

If a(0) is put to 2 instead of 1 this becomes a(n) = (-1)^n*A005248(n), n >= 0. These are then the alternating row sums of triangle A127677.

Also abs(a(n)) is the number of rounded area of pentagon or pentagram in series arrangement. - Kival Ngaokrajang, Mar 27 2013

Colin Barker, Table of n, a(n) for n = 0..1000
Richard M. Low and Ardak Kapbasov, Non-Attacking Bishop and King Positions on Regular and Cylindrical Chessboards, Journal of Integer Sequences, Vol. 20 (2017), Article 17.6.1, Table 8.
Kival Ngaokrajang, Pentagram for n = 1..6
Eric Weisstein's World of Mathematics, Pentagram
Index entries for linear recurrences with constant coefficients, signature (-3,-1).

Cf. A099837 (row sums of A110162).

Cf. A000032, A000045, A001254, A005248, A075150, A127677.

A219233:= func< n | n eq 0 select 1 else (-1)^n*Lucas(2*n) >; // G. C. Greubel, Jun 13 2025

A219233[n_]:= (-1)^n*LucasL[2*n] - Boole[n==0]; (* G. C. Greubel, Jun 13 2025 *)

Vec((1-x^2)/(1+3*x+x^2) + O(x^40)) \\ Colin Barker, Oct 14 2015

def A219233(n): return (-1)**n*lucas_number2(2*n,1,-1) - int(n==0) # G. C. Greubel, Jun 13 2025

a(0) = 1 and a(n) = (-1)^n*(F(2*(n+1)) - F(2*(n-1))) = (-1)^n*L(2*n), n>=1, with F=A000045 (Fibonacci) and L=A000032 (Lucas).
O.g.f.: (1-x^2)/(1+3*x+x^2).
G.f.: (W(0) -6)/(5*x) -1 , where W(k) = 5*x*k + x + 6 - 6*x*(5*k-9)/W(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Aug 19 2013
From Colin Barker, Oct 14 2015: (Start)
a(n) = -3*a(n-1) - a(n-2) for n>2.
a(n) = (1/2*(-3-sqrt(5)))^n + (1/2*(-3+sqrt(5)))^n for n>0. (End)
E.g.f.: 2*exp(-3*x/2)*cosh(sqrt(5)*x/2) - 1. - Stefano Spezia, Dec 26 2021
From G. C. Greubel, Jun 13 2025: (Start)
a(-n) = a(n).
a(n) = (-1)^n*A001254(n) - 2 - [n=0] = A075150(n) - 2 - [n=0]. (End)

A228783 Table of coefficients of the algebraic number s(2l) = 2sin(Pi/2l) as a polynomial in powers of rho(2l) = 2cos(Pi/(2l)) if l is even and of rho(l) = 2*cos(Pi/l) if l is odd (reduced version).

Original entry on oeis.org

2, 0, 1, 1, 0, -3, 0, 1, -1, 1, 0, 4, 0, -1, -1, -1, 1, 0, -7, 0, 14, 0, -7, 0, 1, 2, 1, -1, 0, 8, 0, -18, 0, 8, 0, -1, 1, 2, -3, -1, 1, 0, -8, 0, 6, 0, -1, 0, 0, -1, 3, 3, -4, -1, 1, 0, 12, 0, -67, 0, 96, 0, -52, 0, 12, 0, -1, -2, 3, 1, -1, 0, -15, 0, 140, 0, -378, 0, 450, 0, -275, 0, 90, 0, -15, 0, 1

Offset: 1

Wolfdieter Lang, Oct 06 2013

In the regular (2*l)-gon inscribed in a circle of radius R the length ratio side/R is s(2*l) = 2*sin(Pi/(2*l)). This can be written as a polynomial in the length ratio (smallest diagonal)/side given by rho(2*l) = 2*cos(Pi/(2*l)). (For the 2-gon there is no such diagonal and rho(2) = 0). This leads, in a first step, to the triangle A127672 (see the Oct 05 2013 comment there referring also to the bisections signed A111125 and A127677). Because the minimal polynomial of the algebraic number rho(2*l) of degree delta(2*l) = A055034(2*l), called C(2*l,x) (with coefficients given in A187360) one can eliminate all powers rho(2*l)^k with k >= delta(2*l) by using C(2*l,rho(2*l)) = 0. Furthermore, because for odd l only even powers of rho(2*l) appear, but rho(2*l)^2 = 2 + rho(l), one will obtain a reduced table for s(2*l) with powers rho(2*l)^(2*k+1), k= 0, ..., (delta(2*l)-2)/2 if l is even, and with powers rho(l)^m, m=0, ... , delta(l)-1 if l is odd.
This leads to a reduction of the triangle A127672, when applied for the s(2*l) computation, giving the present table with row length delta(4*L) = A055034(4*L) = phi(8*L)/2 if l =2*L, if L >= 1, and  phi(2*L+1)/2 = A055035(4*L+2), if l = 2*L + 1,  L >= 1, where phi(n) = A000010(n) (Euler totient).
This table gives the coefficients of s(2*l) in the power basis of the algebraic number field Q(rho(2*l)) of degree delta(2*l) = A055034(2*l) if l is even, and in Q(rho(l)) of degree delta(2*l)/2 if l is odd. s(2) and s(6) are rational integers of degree 1.
Thanks go to Seppo Mustonen whose question about the square of the sum of all length in a regular n-gon, led me to this computation.
If l = 2*L+1, L >= 0, that is n == 2 (mod 4), one can obtain s(2*l) more directly in powers of rho(l) by s(2*l) = R(l-1, rho(l)) (mod C(l,rho(l))), with the monic (except for l=1) Chebyshev T-polynomials, called R, in A127672, and the C polynomials from A187360. - Wolfdieter Lang, Oct 10 2013

			The table a(l,m), with n = 2*l, begins:
n,  l \m  0   1   2    3   4   5   6    7   8   9  10  11 ...
2   1:    2
4   2:    0   1
6   3:    1
8   4:    0  -3   0    1
10  5:   -1   1
12  6:    0   4   0   -1
14  7:   -1  -1   1
16  8:    0  -7   0   14   0  -7   0    1
18  9:    2   1  -1
20 10:    0   8   0  -18   0   8   0   -1
22 11:    1   2  -3   -1   1
24 12:    0  -8   0    6   0  -1   0    0
26 13:   -1   3   3   -4  -1   1
28 14:    0  12   0  -67   0  96   0  -52  0  12  0  -1
30 15:   -2   3   1   -1
...
n = 8, l = 4:  s(8)  = -3*rho(8) + rho(8)^3 = -3*sqrt(2 + sqrt(2)) + (sqrt(2 + sqrt(2)))^3 = (sqrt(2) - 1)*sqrt(2 + sqrt(2)).
n = 10, l = 5:  s(10) =  -1 + rho(5), where rho(5) = tau = (1 + sqrt(5))/2, the golden section.

Cf. A127672, A111125, A127677, A055034, A187360, A228785 (odd n case), A228786 (minimal polynomials).

a(2*L,m) = [x^m](s(4*L,x)(mod C(4*L,x))), with s(4*L,x) = sum((-1)^(L-1-s)*A111125(L-1,s)*x^(2*s+1),s=0..L-1), L >= 1, m =0, ..., delta(4*L)-1, and

a(2*L+1,m) = [x^m](s(4*L+2,x)(mod C(2*L+1,x))), with s(4*L+2,x) = sum(A127677(L,s)*(2+x)^(L-s)),s=0..L) (with s(2,x) = 2 for L = 0), L >= 0, m = 0, ..., delta(4*L+2)/2, with delta(n) = A055034(2*l).

A335391 Square array read by antidiagonals downwards: for n >= 2, T(k,n) is the number of permutations of [k+n] that differ in every position from both the identity permutation and a permutation consisting of k 1-cycles and one n-cycle.

Original entry on oeis.org

2, -1, 0, 0, 1, 2, 1, 0, 1, 4, 2, 3, 4, 7, 18, 13, 16, 19, 24, 35, 88, 80, 95, 114, 137, 168, 221, 530, 579, 672, 783, 916, 1077, 1280, 1589, 3708, 4738, 5397, 6164, 7061, 8114, 9359, 10860, 12979, 29666, 43387, 48704, 54773, 61720, 69697, 78888, 89527, 101976, 118663, 266992

Offset: 0

William P. Orrick, Jun 04 2020

The number of permutations of [k+n] that differ in every position from both the identity permutation and a permutation consisting of k 1-cycles and s cycles of lengths p_1, p_2, ... p_s, with p_j >= 2 and p_1+p_2+...+p_s = n, can be expressed as Sum T(k,p_1+-p_2+-...+-p_s), where the sum is over all 2^(s-1) choices of sign and where T(k,-n) = T(k,n) (Touchard).
The first of Touchard's formulas for T(k,n) involves A034807, the number of k-matchings of C_n (A213234 or A127677 with sign included) and A047920, the k-th differences of the factorial numbers.
A slightly different formula, due to Wyman and Moser in the k=0 case, involves A213234 and A000023.
The first column is twice A000166 (twice the number of derangements of [k]); the second column is A105926 (first differences of A000166); the third column is A331007 (with offset 2); the first row is A102761 (the ménage numbers); the second row is A000270.

			Array starts:
k/n |    0     1      2      3       4         5          6           7
-----------------------------------------------------------------------
0   |    2    -1      0      1       2        13         80         579
1   |    0     1      0      3      16        95        672        5397
2   |    2     1      4     19     114       783       6164       54773
3   |    4     7     24    137     916      7061      61720      602955
4   |   18    35    168   1077    8114     69697     671736     7172007
5   |   88   221   1280   9359   78888    749547    7913440    91815601
6   |  530  1589  10860  89527  837794   8741875  100478588  1260186153
7   | 3708 12979 101976 938181 9669196 110058257 1369406616 18475560567
There are T(1,3)=3 permutations that differ from 1234=(1)(2)(3)(4) and 1342=(1)(234) in every position: 2413, 3421, and 4123.

Mathematics Stack Exchange, Double derangement and the other unfamous kind of derangement.
J. Touchard, Sur un problème de permutations, Comptes Rendus Acad. Sci. Paris, 198 (1934) 631-633.
Max Wyman and Leo Moser, On the problème des ménages, Canadian Journal of Mathematics, 10 (1958) 468-480.

Cf. A034807, A213234, A127677, A047920, A000023, A000166, A105926, A331007, A102761, A000270.

T := proc(n,k) local t; t := proc(n, k) option remember;
   simplify((n + k)!*hypergeom([-n], [-n - k], -1)) end:
   if k = 0 then return 2*t(n, 0) fi;
   add((-1)^j*(2*k)/(2*k-j)*binomial(2*k-j, j)*t(n, k-j), j=0 ..k) end:
seq(lprint(seq(T(n, k), k=0..7)),n=0..7); # Peter Luschny, Jul 22 2020

f(k, n) = sum(j=0, k, (-1)^j*binomial(k, j)*(n+k-j)!);
T(k, n) = if (n==0, 2*f(k, 0), sum(j=0, n, (-1)^j*(2*n)/(2*n-j)*binomial(2*n-j, j)*f(k, n-j)));
matrix(7, 7,n, k, T(n-1,k-1))
\\ Michel Marcus, Jun 26 2020

def f(k,n):
    return sum((-1)^j*binomial(k,j)*factorial(n+k-j) for j in range(0,k+1))
def T(k,n):
    if n==0:
        return 2*f(k,0)
    else:
        return sum((-1)^j*(2*n)/(2*n-j)*binomial(2*n-j,j)*f(k,n-j) for j in range(0,n+1))

T(k,0) = 2*nu(k,k), T(k,n>0) = Sum_{j=0..n} A213234(2*n,j)*nu(k,k+n-j) = Sum_{j=0..n} (-1)^j*2*n/(2*n-j)*binomial(2*n-j,j)*nu(k,k+n-j) where nu(k,k+n) = A047920(k+n,k) = Sum_{j=0..k} (-1)^j*binomial(k,j)*(k+n-j)! (Touchard).
T(k,n) = 2*cos(2*n * arccos(1/2*sqrt(x))) = 2*Chebyshev_T(2*n,sqrt(x)/2), where, after expanding in powers of x, x^m gets replaced by nu(k,k+m) (Touchard).
T(k,n) = 2*(-1)^n*Sum_{j=0..n} (-1)^j*(Product_{r=0..j} n^2-r^2)/(2*j)!*nu(k,k+j) (Touchard).
T(k,n) = 2*Integral_{x=0..oo} e^(-x^2) * (x^2-1)^k * x * ((x+sqrt(x^2-4))^(2*n)+(x-sqrt(x^2-4))^(2*n)) / 2^(2*n) dx (Touchard).
T(k,0) = 2*Sum_{j=0..h} binomial(h,j)*k(j), T(k,n) = Sum_{i>=0} A213234(n,i)*Sum_{j=0..h} binomial(h,j)*k(n-2*i+j) = Sum_{i>=0} (-1)^i*n/(n-i)*binomial(n-i,i)*Sum_{j=0..h} binomial(h,j)*k(n-2*i+j) where k(n) = A000023(n) = n! * Sum_{i=0..n} (-2)^i / i! (k=0 case due to Wyman and Moser)
T(k+1,n+1) = T(k,n)+T(k,n+1)+T(k,n+2): This holds for all integers n if one defines T(k,-n) = T(k,n).
T(k,0) = 2*A000166(k).
T(k,1) = A105926(k).
T(k,2) = A331007(k+2).
T(0,n) = A102761(n).
T(1,n) = A000270(n).

A152060 Triangle read by rows, characteristic polynomials of Cartan ring matrices.

Original entry on oeis.org

1, 1, -2, 1, -4, 3, 1, -6, 9, -4, 1, -8, 20, -16, 4, 1, -10, 35, -50, 25, -4, 1, -12, 54, -112, 105, -36, 4, 1, -14, 77, -210, 294, -196, 49, -4, 1, -16, 104, -352, 660, -672, 336, -64, 4, 1, -18, 135, -546, 1287, -1782, 1386, -540, 81, -4, 1, -20, 170, -800, 2275, -4004, 4290, -2640, 825, -100, 4

Offset: 0

Gary W. Adamson & Roger L. Bagula, Nov 22 2008

			Triangle begins:
1;
1, -2;
1, -4, 3;
1, -6, 9, -4;
1, -8, 20, -16, 4;
1, -10, 35, -50, 25, -4;
1, -12, 54, -112, 105, -36, 4;
1, -14, 77, -210, 294, -196, 49, -4;
1, -16, 104, -352, 660, -672, 336, -64, 4;
1, -18, 135, -546, 1287, -1782, 1386, -540, 81, -4;
1, -20, 170, -800, 2275, -4004, 4290, -2640, 825, -100, 4;
...
Example: x^5 -10x^4 + 35x^3 -50x^2 + 25x - 4 = (x - 4) * (x^2 - 3x + 1)^2 is the characteristic polynomial of the matrix
[ 2,-1, 0, 0, 1]
[-1, 2,-1, 0, 0]
[ 0,-1, 2,-1, 0]
[ 0, 0,-1, 2,-1]
[ 1, 0, 0,-1, 2].

William G. Harter, University of Arkansas; personal communication

P. Damianou , On the characteristic polynomials of Cartan matrices and Chebyshev polynomials, arXiv preprint arXiv:1110.6620 [math.RT], 2014.
P. Damianou and C. Evripidou, Characteristic and Coxeter polynomials for affine Lie algebras, arXiv preprint arXiv:1409.3956 [math.RT], 2014.
Todd Rowland, Eric Weisstein's World of Mathematics, Cartan Matrix

Cf. A008310.

Cf. A156308, A127677, A217476, A263916.

M[n_] := SparseArray[{Band[{1, 1}] -> 2, Band[{1, 2}] -> -1, Band[{2, 1}] -> -1, {1, n} -> 1, {n, 1} -> 1}, {n, n}];
row[0] = {1}; row[1] = {1, -2};
row[n_] := (-1)^n CharacteristicPolynomial[M[n], x] // CoefficientList[#, x]& // Reverse;
Table[row[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Aug 08 2018 *)

Triangle read by rows, n-th row = characteristic polynomial of n X n Cartan ring matrix, defined as a Cartan matrix with 1's in the upper right and lower left corners, i.e., positions (1,n) and (n,1).
The coefficients of characteristic polynomials of matrices C_n, defined by
C_n=
(2 -1 0 ... 0 1)
(-1 2 -1 0 ... 0)
(0 -1 2 -1 0 ... 0)
...
(0 ... 0 -1 2 -1)
(1 0 ... 0 -1 2),
give the same triangle T(n,k), for n>0, k=0,...,n, with T(0,0)=1. - L. Edson Jeffery, Mar 27 2011
It appears that for n >= 3 the n-th row polynomial equals 2*T(2*n,sqrt(x)/2) + 2*(-1)^n, where T(n,x) denotes the Chebyshev polynomial of the first kind (A008310). Checked for n = 3 through n = 12. - Peter Bala, May 04 2014
Apparently, omitting the diagonal here, this triangular array is signed, reversed A156308 (cf. A127677, A217476, A263916). For relations among the characteristic polynomials of Cartan matrices of the Coxeter root groups, Chebyshev polynomials, cyclotomic polynomials, and the polynomials of this entry, see Damianou (p. 12, 20, and 21) and Damianou and Evripidou (p. 7). - Tom Copeland, Nov 07 2015

Edited by L. Edson Jeffery, Mar 26 2011

Some terms corrected from Peter Bala, May 04 2014

A284966 Triangle read by rows: coefficients of the scaled Lucas polynomials x^(n/2)*L(n, sqrt(x)) for n >= 0, sorted by descending powers of x.

Original entry on oeis.org

2, 1, 0, 2, 1, 0, 0, 3, 1, 0, 0, 2, 4, 1, 0, 0, 0, 5, 5, 1, 0, 0, 0, 2, 9, 6, 1, 0, 0, 0, 0, 7, 14, 7, 1, 0, 0, 0, 0, 2, 16, 20, 8, 1, 0, 0, 0, 0, 0, 9, 30, 27, 9, 1, 0, 0, 0, 0, 0, 2, 25, 50, 35, 10, 1, 0, 0, 0, 0, 0, 0, 11, 55, 77, 44, 11, 1, 0, 0, 0, 0, 0, 0, 2, 36, 105, 112, 54, 12, 1

Offset: 0

Eric W. Weisstein, Apr 06 2017

For n >= 3, also the coefficients of the edge and vertex cover polynomials for the n-cycle graph C_n.

For more information on how this triangular array is related to the work of Charalambides (1991) and Moser and Abramson (1969), see the comments for triangular array A212634 (which contains additional formulas). The coefficients of these polynomials are given by formula (2.1), p. 291, in Charalambides (1991), where an obvious typo in the index of the summation must be corrected (floor(n/K) -> floor(n/K) - 1). - Petros Hadjicostas, Jan 27 2019

			First few polynomials are
  2;
  x;
  2*x + x^2;
  3*x^2 + x^3;
  2*x^2 + 4*x^3 + x^4;
  5*x^3 + 5*x^4 + x^5;
  ...
giving
  2;
  0, 1;
  0, 2, 1;
  0, 0, 3, 1;
  0, 0, 2, 4, 1;
  0, 0, 0, 5, 5, 1;
  ...

C. A. Charalambides, Lucas numbers and polynomials of order k and the length of the longest circular success run, The Fibonacci Quarterly, 29 (1991), 290-297.
W. O. J. Moser and M. Abramson, Enumeration of combinations with restricted differences and cospan, J. Combin. Theory, 7 (1969), 162-170.
Eric Weisstein's World of Mathematics, Cycle Graph
Eric Weisstein's World of Mathematics, Edge Cover Polynomial
Eric Weisstein's World of Mathematics, Lucas Polynomial
Eric Weisstein's World of Mathematics, Vertex Cover Polynomial

Cf. A034807 (Lucas polynomials x^(n/2)*L(n, 1/sqrt(x))).

Cf. A111125, A127677, A136481, A212634.

L := proc (n, K, x) -1 + sum((-1)^j*n*binomial(n - j*K, j)*x^j*(x+1)^(n - j*(K+1))/(n - j*K), j = 0 .. floor(n/(K + 1))) end proc; for i to 30 do expand(L(i, 2, x)) end do; # gives the g.f. of row n for 1 <= n <= 30. - Petros Hadjicostas, Jan 27 2019

CoefficientList[Table[x^(n/2) LucasL[n, Sqrt[x]], {n, 12}], x] // Flatten (* Eric W. Weisstein, Apr 06 2017 *)
CoefficientList[Table[2 x^n (-1/x)^(n/2) ChebyshevT[n, 1/(2 Sqrt[-1/x])], {n, 12}], x] // Flatten (* Eric W. Weisstein, Apr 06 2017 *)
CoefficientList[Table[FunctionExpand[2 (-(1/x))^(n/2) x^n Cos[n ArcSec[2 Sqrt[-(1/x)]]]], {n, 15}], x] // Flatten (* Eric W. Weisstein, Apr 06 2017 *)
CoefficientList[LinearRecurrence[{x, x}, {x, x (2 + x)}, 15], x] // Flatten (* Eric W. Weisstein, Apr 06 2017 *)

First element T(n=0, k=0) and the example corrected by Petros Hadjicostas, Jan 27 2019

Name edited by Petros Hadjicostas, Jan 27 2019 and by Stefano Spezia, Mar 09 2025

A284982 Triangle read by rows: coefficients of the Laplacian polynomial of the n-cycle graph C_n.

Original entry on oeis.org

0, -1, 0, -4, 1, 0, -9, 6, -1, 0, -16, 20, -8, 1, 0, -25, 50, -35, 10, -1, 0, -36, 105, -112, 54, -12, 1, 0, -49, 196, -294, 210, -77, 14, -1, 0, -64, 336, -672, 660, -352, 104, -16, 1, 0, -81, 540, -1386, 1782, -1287, 546, -135, 18, -1, 0, -100, 825, -2640, 4290, -4004, 2275, -800, 170, -20, 1

Offset: 1

Eric W. Weisstein, Apr 06 2017

Extended to n = 1 and n = 2 using the closed form.

Closely related to A127677, which has opposite signs and rows begin with 2 of alternating signs instead of 0.

			First few polynomials are
-x
-4*x + x^2
-9*x + 6*x^2 - x^3
which give the triangle of coefficients:
0, -1;
0, -4, 1;
0, -9, 6, -1;
...

Eric Weisstein's World of Mathematics, Cycle Graph
Eric Weisstein's World of Mathematics, Laplacian Polynomial

Cf. A127677 (opposite signs and rows begin with +/2).

CoefficientList[LucasL[2 Range[10], Sqrt[-x]] - 2, x] // Flatten (* Eric W. Weisstein, Apr 06 2017 *)
CoefficientList[2 (ChebyshevT[Range[10], 1 - x/2] - 1), x] // Flatten (* Eric W. Weisstein, Apr 06 2017 *)
CoefficientList[FunctionExpand[2 (Cos[Range[10] ArcCos[1 - x/2]] - 1)], x] // Flatten (* Eric W. Weisstein, Apr 06 2017 *)
CoefficientList[LinearRecurrence[{3 - x, -3 + x, 1}, {-x, (-4 + x) x, -(-3 + x)^2 x}, 10], x] // Flatten (* Eric W. Weisstein, Apr 06 2017 *)

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A084534 Triangle read by rows: row #n has n+1 terms. T(n,0)=1, T(n,n)=2, T(n,m) = T(n-1,m-1) + Sum_{k=0..m} T(n-1-k,m-k).

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A219233 Alternating row sums of Riordan triangle A110162.

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A228783 Table of coefficients of the algebraic number s(2*l) = 2*sin(Pi/2*l) as a polynomial in powers of rho(2*l) = 2*cos(Pi/(2*l)) if l is even and of rho(l) = 2*cos(Pi/l) if l is odd (reduced version).

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A335391 Square array read by antidiagonals downwards: for n >= 2, T(k,n) is the number of permutations of [k+n] that differ in every position from both the identity permutation and a permutation consisting of k 1-cycles and one n-cycle.

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A152060 Triangle read by rows, characteristic polynomials of Cartan ring matrices.

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A284966 Triangle read by rows: coefficients of the scaled Lucas polynomials x^(n/2)*L(n, sqrt(x)) for n >= 0, sorted by descending powers of x.

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A284982 Triangle read by rows: coefficients of the Laplacian polynomial of the n-cycle graph C_n.

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A228783 Table of coefficients of the algebraic number s(2l) = 2sin(Pi/2l) as a polynomial in powers of rho(2l) = 2cos(Pi/(2l)) if l is even and of rho(l) = 2*cos(Pi/l) if l is odd (reduced version).