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A305721
Crystal ball sequence for the lattice C_7.
2
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
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.
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
Sequence in context: A259693 A196785 A196808 * A197372 A174944 A221330
KEYWORD
nonn,easy
AUTHOR
Seiichi Manyama, Jun 09 2018
STATUS
approved