A305721 Crystal ball sequence for the lattice C_7.

1, 99, 1765, 14407, 74313, 284075, 880685, 2340495, 5529233, 11905267, 23784309, 44673751, 79684825, 136030779, 223619261, 355747103, 549905697, 828705155, 1220925445, 1762702695, 2498858857, 3484382923, 4786071885, 6484339631, 8675201969, 11472445971, 15009991829

Offset

0,2

Comments

Partial sums of A019563.

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 (8,-28,56,-70,56,-28,8,-1).

Formula

a(n) = 8*a(n-1) - 28*a(n-2) + 56*a(n-3) - 70*a(n-4) + 56*a(n-5) - 28*a(n-6) + 8*a(n-7) - a(n-8), for n>7.

a(n) = Sum_{k=0..7} binomial(14, 2k)*binomial(n+k, 7).

G.f.: (1 + x)*(1 + 90*x + 911*x^2 + 2092*x^3 + 911*x^4 + 90*x^5 + x^6) / (1 - x)^8. - Colin Barker, Jun 09 2018

Mathematica

Array[Sum[Binomial[14, 2 k] Binomial[# + k, 7], {k, 0, 7}] &, 27, 0] (* Michael De Vlieger, Jun 11 2018 *)
LinearRecurrence[{8, -28, 56, -70, 56, -28, 8, -1}, {1, 99, 1765, 14407, 74313, 284075, 880685, 2340495}, 30] (* Harvey P. Dale, May 16 2023 *)

Prog

(PARI) {a(n) = sum(k=0, 7, binomial(14, 2*k)*binomial(n+k, 7))}
(PARI) Vec((1 + x)*(1 + 90*x + 911*x^2 + 2092*x^3 + 911*x^4 + 90*x^5 + x^6) / (1 - x)^8 + O(x^40)) \\ Colin Barker, Jun 09 2018
(GAP) b:=7;; List([0..30], n->Sum([0..b], k->Binomial(2*b, 2*k)*Binomial(n+k, b))); # Muniru A Asiru, Jun 09 2018

Crossrefs

Cf. A019563, A142992.

Sequence in context: A259693 A196785 A196808 * A197372 A174944 A221330

Adjacent sequences: A305718 A305719 A305720 * A305722 A305723 A305724

Keyword

nonn,easy

Author

Seiichi Manyama, Jun 09 2018

Status

approved