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A305723
Crystal ball sequence for the lattice C_9.
2
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
0,2
COMMENTS
Partial sums of A035746.
LINKS
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).
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)
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 *)
PROG
(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
(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
CROSSREFS
Sequence in context: A109343 A217644 A185500 * A232260 A027543 A222837
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Jun 09 2018
STATUS
approved