A305724 Crystal ball sequence for the lattice C_10.

1, 201, 7001, 104881, 927441, 5707449, 26986089, 104535009, 346615329, 1014889769, 2684641785, 6526963345, 14778775025, 31490462745, 63670078985, 122977987009, 228167048769, 408511495049, 708522994329, 1194315679089, 1962053519121, 3148993975161

Offset: 0

Author

Seiichi Manyama, Jun 09 2018

Keywords

• nonn

Comments

Partial sums of A035747.

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 (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Crossrefs

Cf. A035747, A142992.

Programs

GAP

b:=10;; 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, 10, binomial(20, 2*k)*binomial(n+k, 10))}

PARI

Vec((1 + 6*x + x^2)*(1 + 184*x + 3740*x^2 + 16136*x^3 + 25414*x^4 + 16136*x^5 + 3740*x^6 + 184*x^7 + x^8) / (1 - x)^11 + O(x^30)) \\ Colin Barker, Jun 09 2018

Formula

a(n) = 11*a(n-1) - 55*a(n-2) + 165*a(n-3) - 330*a(n-4) + 462*a(n-5) - 462*a(n-6) + 330*a(n-7) - 165*a(n-8) + 55*a(n-9) - 11*a(n-10) + a(n-11) for n>10.

a(n) = Sum_{k=0..10} binomial(20, 2k)*binomial(n+k, 10).

G.f.: (1 + 6*x + x^2)*(1 + 184*x + 3740*x^2 + 16136*x^3 + 25414*x^4 + 16136*x^5 + 3740*x^6 + 184*x^7 + x^8) / (1 - x)^11. - Colin Barker, Jun 09 2018