A305722 Crystal ball sequence for the lattice C_8.

1, 129, 2945, 29953, 187137, 845185, 3032705, 9173505, 24331777, 58161793, 127791489, 261902081, 506298625, 931299201, 1641303169, 2786931713, 4580166657, 7312946305, 11379709825, 17304414465, 25772582657, 37668968833, 54121468545, 76551925249

Offset: 0

Author

Seiichi Manyama, Jun 09 2018

Keywords

• nonn

Comments

Partial sums of A019564.

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 (9,-36,84,-126,126,-84,36,-9,1).

Crossrefs

Cf. A019564, A142992.

Programs

GAP

b:=8;; List([0..25],n->Sum([0..b],k->Binomial(2*b,2*k)*Binomial(n+k,b))); # Muniru A Asiru, Jun 09 2018

PARI

{a(n) = sum(k=0, 8, binomial(16, 2*k)*binomial(n+k, 8))}

PARI

Vec((1 + 120*x + 1820*x^2 + 8008*x^3 + 12870*x^4 + 8008*x^5 + 1820*x^6 + 120*x^7 + x^8) / (1 - x)^9 + O(x^40)) \\ Colin Barker, Jun 09 2018

Formula

a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + a(n-9), for n>8.

a(n) = Sum_{k=0..8} binomial(16, 2k)*binomial(n+k, 8).

G.f.: (1 + 120*x + 1820*x^2 + 8008*x^3 + 12870*x^4 + 8008*x^5 + 1820*x^6 + 120*x^7 + x^8) / (1 - x)^9. - Colin Barker, Jun 09 2018