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A142994 Crystal ball sequence for the lattice C_5. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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.

LINKS

Table of n, a(n) for n=0..28.

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).

FORMULA

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(10,2k)*binomial(n+k,5).

a(n) = Sum_{k = 0..5} binomial(10,2k+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

EXAMPLE

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 sequence begins 1, 1 + 50 = 51, ... .

MAPLE

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

MATHEMATICA

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 *)

PROG

(Python)

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

(MAGMA) [(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

CROSSREFS

Cf. A019561, A063496, A142992, A142993.

Sequence in context: A204215 A164646 A128511 * A251932 A173804 A166820

Adjacent sequences:  A142991 A142992 A142993 * A142995 A142996 A142997

KEYWORD

nonn,easy

AUTHOR

Peter Bala, Jul 18 2008

STATUS

approved

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Last modified February 23 21:23 EST 2018. Contains 299588 sequences. (Running on oeis4.)