A305549 Crystal ball sequence for the lattice C_6.

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

Author

Seiichi Manyama, Jun 09 2018

Keywords

• nonn
• easy

Comments

Partial sums of A019562.

Links

• 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).

Crossrefs

Cf. A019562, A142992.

Programs

PARI

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

PARI

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

Formula

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)