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, pp. 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, 2*k)*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

From Peter Bala, Mar 12 2024: (Start)

Sum_{k >= 1} 1/(k*a(k)*a(k-1)) = 2*ln(2) - 289/210 = 1/(99 - 3/(107 - 60/(123 - 315/(147 - ... - n^2*(4*n^2 - 1)/((2*n + 1)^2 + 2*7^2 - ...))))).

E.g.f.: exp(x)*(1 + 98*x + 1568*x^2/2! + 9408*x^3/3! + 26880*x^4/4! + 39424*x^5/5! + 28672*x^6/6! + 8192*x^7/7!).

Note that -T(14, i*sqrt(x)) = 1 + 98*x + 1568*x^2 + 9408*x^3 + 26880*x^4 + 39424*x^5 + 28672*x^6 + 8192*x^7, where T(n, x) denotes the n-th Chebyshev polynomial of the first kind. See A008310.

Row 7 of A142992. (End)

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