A305723 Crystal ball sequence for the lattice C_9.

1, 163, 4645, 57799, 432073, 2286955, 9446125, 32398735, 96220561, 254831667, 614859189, 1373356887, 2874747225, 5693596923, 10751213181, 19475555103, 34015593249, 57523019715, 94516111685, 151342583015, 236760421097, 362658000011, 544937185805, 804585705647

Offset: 0

Author

Seiichi Manyama, Jun 09 2018

Keywords

• nonn

Comments

Partial sums of A035746.

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 (10,-45,120,-210,252,-210,120,-45,10,-1).

Crossrefs

Cf. A035746, A142992.

Programs

GAP

b:=9;; List([0..25],n->Sum([0..b],k->Binomial(2*b,2*k)*Binomial(n+k,b))); # Muniru A Asiru, Jun 09 2018

Mathematica

Table[Sum[Binomial[18,2k]Binomial[n+k,9],{k,0,9}],{n,0,40}] (* or *) LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{1,163,4645,57799,432073,2286955,9446125,32398735,96220561,254831667},40] (* Harvey P. Dale, Jun 09 2023 *)

PARI

{a(n) = sum(k=0, 9, binomial(18, 2*k)*binomial(n+k, 9))}

PARI

Vec((1 + x)*(1 + 14*x + x^2)*(1 + 138*x + 975*x^2 + 1868*x^3 + 975*x^4 + 138*x^5 + x^6) / (1 - x)^10 + O(x^40)) \\ Colin Barker, Jun 09 2018

Formula

a(n) = Sum_{k=0..9} binomial(18, 2k)*binomial(n+k, 9).

From Colin Barker, Jun 09 2018: (Start)

G.f.: (1 + x)*(1 + 14*x + x^2)*(1 + 138*x + 975*x^2 + 1868*x^3 + 975*x^4 + 138*x^5 + x^6) / (1 - x)^10.

a(n) = 10*a(n-1) - 45*a(n-2) + 120*a(n-3) - 210*a(n-4) + 252*a(n-5) - 210*a(n-6) + 120*a(n-7) - 45*a(n-8) + 10*a(n-9) - a(n-10) for n>9.

(End)