A008310 -id:A008310 - cp's OEIS frontend

A370260 a(n) = sqrt(A370259(2*n+1)).

1, 3, 31, 617, 18529, 748859, 38149567, 2348482961, 169641143873, 14071599763379, 1318414335714015, 137720427724123513, 15871136311527376801, 2000355821099358166891, 273735526097742996298111, 40419227378551955037029921, 6405616571975691389276400257

Offset: 0

Peter Bala, Mar 11 2024

The sequence is conjectured to be integral [added 03 Mar 2024: now confirmed - see the Formula section].

Paolo Xausa, Table of n, a(n) for n = 0..300

Cf. A008310, A342205, A370259, A370261, A370262.

A370259 := n -> simplify( (ChebyshevT(n, n+1) - 1)/n^3 ):
seq(sqrt(A370259(2*n+1)), n = 0..20);

Table[Sqrt[(ChebyshevT[k, k + 1] - 1)/k^3], {k, 1, 40, 2}] (* Paolo Xausa, Jul 24 2024 *)

a(n) = sqrt( (T(2*n+1, 2*n+2) - 1)/(2*n+1)^3 ), where T(n, x) denotes the n-th Chebyshev polynomial of the first kind.
a(n) = sqrt( Sum_{k = 1..2*n+1} (2^k)*(2*n + 1)^(k-2)*binomial(2*n + k + 1, 2*k)/(2*n + k + 1) ).
a(n) = Sum_{k = 0..n} binomial(n+k, n-k)/(2*k + 1) * (4*n + 2)^k (shows the sequence to be integral) = R(n,2), where R(n, x) is the n-th row polynomial of A370262. - Peter Bala, Apr 03 2024

A142994 Crystal ball sequence for the lattice C_5.

Original entry on oeis.org

1, 51, 501, 2471, 8361, 22363, 50973, 103503, 192593, 334723, 550725, 866295, 1312505, 1926315, 2751085, 3837087, 5242017, 7031507, 9279637, 12069447, 15493449, 19654139, 24664509, 30648559, 37741809, 46091811, 55858661, 67215511, 80349081

Offset: 0

Peter Bala, Jul 18 2008

The lattice C_5 consists of all integer lattice points v = (x_1,...,x_5) in Z^5 such that (x_1 + ... + x_5) is even, equipped with the taxicab type norm ||v|| = (1/2) * (|x_1| + ... + |x_5|). The crystal ball sequence of C_5 gives the number of lattice points v in C_5 with ||v|| <= n for n = 0,1,2,3,... [Bacher et al.].

Partial sums of A019561.

			a(1) = 51. The origin has norm 0. The 50 lattice points in Z^5 of norm 1 (as defined above) are +-2*e_i, 1 <= i <= 5 and (+- e_i +- e_j), 1 <= i < j <= 5, where e_1, ... , e_5 denotes the standard basis of Z^5. These 50 vectors form a root system of type C_5. Hence the sequence begins 1, 1 + 50 = 51, ... .

Seiichi Manyama, Table of n, a(n) for n = 0..10000
R. Bacher, P. de la Harpe, and B. Venkov, Séries de croissance et séries d'Ehrhart associées aux réseaux de racines, Annales de l'Institut Fourier, Tome 49 (1999) no. 3, pp. 727-762.
R. Bacher, P. de la Harpe and B. Venkov, Séries de croissance et séries d'Ehrhart associées aux réseaux de racines, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Row 5 of A142992. Cf. A019561, A063496, A142993.

[(2*n+1)*(32*n^4+64*n^3+88*n^2+56*n+15)/15: n in [0..30]]; // Vincenzo Librandi, Dec 16 2015

a := n -> (2*n+1)*(32*n^4+64*n^3+88*n^2+56*n+15)/15: seq(a(n), n = 0..20)

CoefficientList[Series[(1 + 45 x + 210 x^2 + 210 x^3 + 45 x^4 + x^5)/(1 - x)^6, {x, 0, 33}], x] (* or *) LinearRecurrence[{6, -15, 20, -15, 6, -1},{1, 51, 501, 2471, 8361, 22363}, 25] (* Vincenzo Librandi, Dec 16 2015 *)

A142994_list, m = [], [512, -768, 352, -48, 2, 1]
for _ in range(10**2):
    A142994_list.append(m[-1])
    for i in range(5):
        m[i+1] += m[i] # Chai Wah Wu, Dec 15 2015

a(n) = (2*n + 1)*(32*n^4 + 64*n^3 + 88*n^2 + 56*n + 15)/15.
a(n) = Sum_{k = 0..5} binomial(10, 2*k)*binomial(n+k, 5).
a(n) = Sum_{k = 0..5} binomial(10, 2*k+1)*binomial(n+k+1/2, 5).
O.g.f.: (1 + 45*x + 210*x^2 + 210*x^3 + 45*x^4 + x^5)/(1 - x)^6 = 1/(1 - x) * T(5, (1 + x)/(1 - x)), where T(n, x) denotes the Chebyshev polynomial of the first kind.
Sum_{n >= 1} 1/(n*a(n-1)*a(n)) = 2*log(2) - 41/30.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6), for n > 5. - Vincenzo Librandi, Dec 16 2015
From Peter Bala, Mar 11 2024: (Start)
Sum_{k = 1..n+1} 1/(k*a(k)*a(k-1)) = 1/(51 - 3/(59 - 60/(75 - 315/(99 - ... - n^2*(4*n^2 - 1)/((2*n + 1)^2 + 2*5^2))))).
E.g.f.: exp(x)*(1 + 50*x + 400*x^2/2! + 1120*x^3/3! + 1280*x^4/4! + 512*x^5/5!).
Note that -T(10, i*sqrt(x)) = 1 + 50*x + 400*x^2 + 1120*x^3 + 1280*x^4 + 512*x^5. See A008310. (End)

A370261 a(n) = sqrt(A370259(2*n)/(n+1)) for n >= 1.

Original entry on oeis.org

1, 5, 65, 1449, 46561, 1968525, 103565057, 6531391313, 480749649601, 40482981221781, 3840053099665729, 405275779792031225, 47113209228513626017, 5982545638922153790749, 823992221632687352744961, 122360935410018418223907489, 19489013519781051891806113153

Offset: 1

Peter Bala, Mar 11 2024

The sequence is conjectured to be integral.

Paolo Xausa, Table of n, a(n) for n = 1..300

Cf. A008310, A342205, A370259, A370260.

A370259 := n -> simplify( (ChebyshevT(n, n+1) - 1)/n^3 ):
seq(sqrt(A370259(2*n)/(n+1)), n = 1..20);

Table[Sqrt[(ChebyshevT[2*n, 2*n + 1] - 1)/(2*n)^3/(n + 1)], {n, 20}] (* Paolo Xausa, Jul 24 2024 *)

from math import isqrt
from sympy import chebyshevt
def A370261(n): return isqrt((chebyshevt((m:=n<<1),m+1)-1)//((n+1)*m**3)) # Chai Wah Wu, Mar 13 2024

a(n) = sqrt( (T(2*n, 2*n+1) - 1)/((n+1)*(2*n)^3) ), where T(n, x) is the n-th Chebyshev polynomial of the first kind.

A081265 Triangle of coefficients of the polynomials a(n, x) = 2a(n-1, x)+ x^2a(n-2,x), n >= 1, a(0, x) = 1, a(1, x) = 1.

Original entry on oeis.org

1, 1, 0, 2, 0, 1, 4, 0, 3, 0, 8, 0, 8, 0, 1, 16, 0, 20, 0, 5, 0, 32, 0, 48, 0, 18, 0, 1, 64, 0, 112, 0, 56, 0, 7, 0, 128, 0, 256, 0, 160, 0, 32, 0, 1, 256, 0, 576, 0, 432, 0, 120, 0, 9, 0, 512, 0, 1280, 0, 1120, 0, 400, 0, 50, 0, 1, 1024, 0, 2816, 0, 2816, 0, 1232, 0, 220

Offset: 0

Paul Barry, Mar 15 2003

Unsigned Chebyshev numbers of the first kind. Columns include A011782, A001792, A001793, A001794, A006974.

For the Riordan coefficient triangle for Chebyshev's T-polynomials (decreasing odd or even powers of x) see A039991. - Wolfdieter Lang, Aug 06 2014

			Triangle rows are {1}, {1,0}, {2,0,1}, {4,0,3,0}, {8,0,8,0,1},.... [Corrected by _Philippe Deléham_, Dec 27 2007]
See the unsigned example under A039991. - _Wolfdieter Lang_, Aug 06 2014

Cf. A008310, A039991 (signed).

T(n,k) = [x^k] a(n,x), k = 0, 1, ..., n, with polynomial a(n,x) defined by the recurrence given as name. Its Binet-de Moivre form is a(n, x) = ((1+sqrt(x^2+1))^n + (1-sqrt(x^2+1))^n)/2.

O.g.f. for row polynomials a(n,x): (1-z)/(1 - 2*z - (x*z)^2). Compare with A039991.

Edited. Name and formula clarified. G.f. of row polynomial, and crossref. A039991 added. - Wolfdieter Lang, Aug 06 2014

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

A175243 Array read by antidiagonals: total number of spanning trees R_n(m) of the complete prism K_m X C_n.

Original entry on oeis.org

1, 2, 1, 3, 12, 3, 4, 75, 294, 16, 5, 384, 11664, 16384, 125, 6, 1805, 367500, 5647152, 1640250, 1296, 7, 8100, 10609215, 1528823808, 6291456000, 259200000, 16807, 8, 35287, 292626432, 380008339280, 18911429680500, 13556617751088, 59549251454

Offset: 1

R. J. Mathar, Mar 13 2010

			The array starts in row n=1 as:
  1,    1,        3,         16,        125
  2,   12,      294,      16384,    1640250
  3,   75,    11664,    5647152, 6291456000
  4,  384,   367500, 1528823808,
  5, 1805, 10609215,

F. T. Boesch and H. Prodinger, Spanning tree formulas and Chebyshev polynomials, Graphs Combinat. 2 (1986) 191-200.
Index entries for sequences related to trees

Cf. A006235 (column 2), A000272, A212798 (column 3).

A175243 := proc(n,m) n*2^(m-1)/m*( orthopoly[T](n,1+m/2)-1)^(m-1) ; end proc:
for d from 2 to 10 do for m from 1 to d-1 do n := d-m ; printf("%d,",A175243(n,m)) ; end do: end do:

r[n_, m_] := n*2^(m-1)*(ChebyshevT[n, 1+m/2]-1)^(m-1)/m; Table[r[n-m, m], {n, 2, 9}, {m, 1, n-1}] // Flatten (* Jean-François Alcover, Jan 10 2014 *)

R_n(m) = n*2^(m-1)* (T(n,1+m/2)-1)^(m-1)/m, where T(n,x) are Chebyshev polynomials, A008310.

Each column of the array is a linear divisibility sequence. Conjecturally, the k-th column satisfies a linear recurrence of order 4*k - 2. - Peter Bala, May 04 2014

A305549 Crystal ball sequence for the lattice C_6.

Original entry on oeis.org

1, 73, 985, 6321, 26577, 85305, 227305, 528865, 1110049, 2149033, 3898489, 6704017, 11024625, 17455257, 26751369, 39855553, 57926209, 82368265, 114865945, 157417585, 212372497, 282469881, 370879785, 481246113, 617731681, 785065321, 988591033, 1234319185

Offset: 0

Seiichi Manyama, Jun 09 2018

Partial sums of A019562.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
R. Bacher, P. de la Harpe, and B. Venkov, Séries de croissance et séries d'Ehrhart associées aux réseaux de racines, Annales de l'Institut Fourier, Tome 49 (1999) no. 3 , p. 727-762.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A019562, A142992.

{a(n) = sum(k=0, 6, binomial(12,2*k)*binomial(n+k,6))}

Vec((1 + 6*x + x^2)*(1 + 60*x + 134*x^2 + 60*x^3 + x^4) / (1 - x)^7 + O(x^40)) \\ Colin Barker, Jun 09 2018

a(n) = (128*n^6 + 384*n^5 + 800*n^4 + 960*n^3 + 692*n^2 + 276*n + 45)/45.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7), for n > 6.
a(n) = Sum_{k = 0..6} binomial(12, 2*k)*binomial(n+k, 6).
G.f.: (1 + 6*x + x^2)*(1 + 60*x + 134*x^2 + 60*x^3 + x^4) / (1 - x)^7. - Colin Barker, Jun 09 2018
From Peter Bala, Mar 12 2024: (Start)
Sum_{k >= 1} 1/(k*a(k)*a(k-1)) = 7/5 - 2*log(2) = 1/(73 - 3/(81 - 60/(97 - 315/(121 - ... - n^2*(4*n^2 - 1)/((2*n + 1)^2 + 2*6^2 - ...))))).
E.g.f.: exp(x)*(1 + 72*x + 840*x^2/2! + 3584*x^3/3! + 6912*x^4/4! + 6144*x^5/5! + 2048*x^6/6!).
Note that T(12, i*sqrt(x)) = 1 + 72*x + 840*x^2 + 3584*x^3 + 6912*x^4 + 6144*x^5 + 2048*x^6, where T(n, x) denotes the n-th Chebyshev polynomial of the first kind. See A008310.
Row 6 of A142992. (End)

A305721 Crystal ball sequence for the lattice C_7.

Original entry on oeis.org

1, 99, 1765, 14407, 74313, 284075, 880685, 2340495, 5529233, 11905267, 23784309, 44673751, 79684825, 136030779, 223619261, 355747103, 549905697, 828705155, 1220925445, 1762702695, 2498858857, 3484382923, 4786071885, 6484339631, 8675201969, 11472445971, 15009991829

Offset: 0

Seiichi Manyama, Jun 09 2018

Partial sums of A019563.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
R. Bacher, P. de la Harpe and B. Venkov, Séries de croissance et séries d'Ehrhart associées aux réseaux de racines, Annales de l'Institut Fourier, Tome 49 (1999) no. 3, pp. 727-762.
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A019563, A142992.

b:=7;; List([0..30],n->Sum([0..b],k->Binomial(2*b,2*k)*Binomial(n+k,b))); # Muniru A Asiru, Jun 09 2018

Array[Sum[Binomial[14, 2 k] Binomial[# + k, 7], {k, 0, 7}] &, 27, 0] (* Michael De Vlieger, Jun 11 2018 *)
LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{1,99,1765,14407,74313,284075,880685,2340495},30] (* Harvey P. Dale, May 16 2023 *)

{a(n) = sum(k=0, 7, binomial(14, 2*k)*binomial(n+k, 7))}

Vec((1 + x)*(1 + 90*x + 911*x^2 + 2092*x^3 + 911*x^4 + 90*x^5 + x^6) / (1 - x)^8 + O(x^40)) \\ Colin Barker, Jun 09 2018

a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8), for n > 7.
a(n) = Sum_{k = 0..7} binomial(14, 2*k)*binomial(n+k, 7).
G.f.: (1 + x)*(1 + 90*x + 911*x^2 + 2092*x^3 + 911*x^4 + 90*x^5 + x^6) / (1 - x)^8. - Colin Barker, Jun 09 2018
From Peter Bala, Mar 12 2024: (Start)
Sum_{k >= 1} 1/(k*a(k)*a(k-1)) = 2*ln(2) - 289/210 = 1/(99 - 3/(107 - 60/(123 - 315/(147 - ... - n^2*(4*n^2 - 1)/((2*n + 1)^2 + 2*7^2 - ...))))).
E.g.f.: exp(x)*(1 + 98*x + 1568*x^2/2! + 9408*x^3/3! + 26880*x^4/4! + 39424*x^5/5! + 28672*x^6/6! + 8192*x^7/7!).
Note that -T(14, i*sqrt(x)) = 1 + 98*x + 1568*x^2 + 9408*x^3 + 26880*x^4 + 39424*x^5 + 28672*x^6 + 8192*x^7, where T(n, x) denotes the n-th Chebyshev polynomial of the first kind. See A008310.
Row 7 of A142992. (End)

A370262 Triangle read by rows: T(n, k) = binomial(n+k, n-k)/(2k + 1) (2*n + 1)^k.

Original entry on oeis.org

1, 1, 1, 1, 5, 5, 1, 14, 49, 49, 1, 30, 243, 729, 729, 1, 55, 847, 5324, 14641, 14641, 1, 91, 2366, 26364, 142805, 371293, 371293, 1, 140, 5670, 101250, 928125, 4556250, 11390625, 11390625, 1, 204, 12138, 324258, 4593655, 36916282, 168962983, 410338673, 410338673

Offset: 0

Peter Bala, Mar 12 2024

The table entries are integers since a(n, k) := binomial(n+k, n-k)/(2*k + 1) * (2*n + 1) gives the entries of the transpose of triangle A082985.

			Triangle begins
 n\k | 0    1      2       3        4        5        6
 - - - - - - - - - - - - - - - - - - - - - - - - - - - -
   0 | 1
   1 | 1    1
   2 | 1    5      5
   3 | 1   14     49      49
   4 | 1   30    243     729      729
   5 | 1   55    847    5324    14641    14641
   6 | 1   91   2366   26364   142805   371293   371293
  ...

A371697 (row sums), A052750 (main diagonal and subdiagonal), A000330 (column 1).

Cf. A008310, A082985, A258708, A370259, A370260.

seq(seq(binomial(n+k, n-k)/(2*k + 1) * (2*n + 1)^k, k = 0..n), n = 0..10);

Table[Binomial[n + k, n - k] / (2*k + 1) * (2*n + 1)^k, {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Apr 17 2024 *)

n-th row polynomial R(n, x) = Sum_{k = 0..n} T(n, k)*x^k = sqrt( 2* Sum_{k = 0..2*n} (2*n + 1)^(k-1) *binomial(2*n+k+2, 2*k+2)/(2*n + k + 2) * x^k ).
R(n, x)^2 = 2/(x*(2*n + 1)^3) * ( ChebyshevT(2*n+1, 1 + (2*n+1)*x/2) - 1 ).
R(n, 2) = A370260(n).

A136388 Triangle read by rows of coefficients of Chebyshev-like polynomials P_{n,2}(x) with 0 omitted (exponents in increasing order).

Original entry on oeis.org

1, -2, 2, 1, -5, 4, 4, -12, 8, -1, 13, -28, 16, -6, 38, -64, 32, 1, -25, 104, -144, 64, 8, -88, 272, -320, 128, -1, 41, -280, 688, -704, 256, -10, 170, -832, 1696, -1536, 512, 1, -61, 620, -2352, 4096, -3328, 1024, 12, -292, 2072, -6400, 9728, -7168, 2048

Offset: 2

Milan Janjic, Mar 30 2008, entry revised Apr 05 2008

If U_n(x), T_n(x) are Chebyshev's polynomials then U_n(x)=P_{n,0}(x), T_n(x)=P_{n,1}(x).

Let n>=2 and k be of the same parity. Consider a set X consisting of (n+k)/2-2 main blocks of the size 2 and an additional block of the size 2, then (-1)^((n-k)/2)a(n,k) is the number of n-2-subsets of X intersecting each main block.

			Rows are (1),(-2,2),(1,-5,4),(4,-12,8),(-1,13,-28,16),...
since P_{2,2}=x^2, P_{3,2}=-2x+2x^3, P_{4,2}=1-5x^2+4x^4,...

Michael De Vlieger, Table of n, a(n) for n = 2..10199 (rows 2 <= n <= 200, flattened).
Milan Janjic, Two enumerative functions.
M. Janjic, On a class of polynomials with integer coefficients, JIS 11 (2008) 08.5.2.
Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.

Cf. A008310, A053117.

if modp(n-k, 2)=0 then a[n,k]:=(-1)^((n-k)/2)*sum((-1)^i*binomial((n+k)/2-2, i)*binomial(n+k-2-2*i, n-2), i=0..(n+k)/2-2); end if;

Rest@ Flatten@ Table[If[SameQ @@ Mod[{n, k}, 2], (-1)^((n - k)/2)*Sum[(-1)^i*Binomial[(n + k)/2 - 2, i]*Binomial[n + k - 2 - 2 i, n - 2], {i, 0, (n + k)/2 - 2}], 0], {n, 2, 13}, {k, Boole@ OddQ@ n, n, 2}] (* Michael De Vlieger, Jul 02 2019 *)

If n>=2 and k are of the same parity then a(n,k)= (-1)^((n-k)/2)*sum((-1)^i*binomial((n+k)/2-2, i)*binomial(n+k-2-2*i, n-2), i=0..(n+k)/2-2) and a(n,k)=0 if n and k are of different parity.

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A370260 a(n) = sqrt(A370259(2*n+1)).

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A142994 Crystal ball sequence for the lattice C_5.

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A370261 a(n) = sqrt(A370259(2*n)/(n+1)) for n >= 1.

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A081265 Triangle of coefficients of the polynomials a(n, x) = 2*a(n-1, x)+ x^2*a(n-2,x), n >= 1, a(0, x) = 1, a(1, x) = 1.

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

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A175243 Array read by antidiagonals: total number of spanning trees R_n(m) of the complete prism K_m X C_n.

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A305549 Crystal ball sequence for the lattice C_6.

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PARI

PARI

A081265 Triangle of coefficients of the polynomials a(n, x) = 2a(n-1, x)+ x^2a(n-2,x), n >= 1, a(0, x) = 1, a(1, x) = 1.

A370262 Triangle read by rows: T(n, k) = binomial(n+k, n-k)/(2k + 1) (2*n + 1)^k.