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
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).
Programs
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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
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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 *) -
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
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)
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