OFFSET
0,2
COMMENTS
Partial sums of A019562.
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 (7,-21,35,-35,21,-7,1).
FORMULA
a(n) = (128*n^6 + 384*n^5 + 800*n^4 + 960*n^3 + 692*n^2 + 276*n + 45)/45.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7), for n > 6.
a(n) = Sum_{k = 0..6} binomial(12, 2*k)*binomial(n+k, 6).
G.f.: (1 + 6*x + x^2)*(1 + 60*x + 134*x^2 + 60*x^3 + x^4) / (1 - x)^7. - Colin Barker, Jun 09 2018
From Peter Bala, Mar 12 2024: (Start)
Sum_{k >= 1} 1/(k*a(k)*a(k-1)) = 7/5 - 2*log(2) = 1/(73 - 3/(81 - 60/(97 - 315/(121 - ... - n^2*(4*n^2 - 1)/((2*n + 1)^2 + 2*6^2 - ...))))).
E.g.f.: exp(x)*(1 + 72*x + 840*x^2/2! + 3584*x^3/3! + 6912*x^4/4! + 6144*x^5/5! + 2048*x^6/6!).
Note that T(12, i*sqrt(x)) = 1 + 72*x + 840*x^2 + 3584*x^3 + 6912*x^4 + 6144*x^5 + 2048*x^6, where T(n, x) denotes the n-th Chebyshev polynomial of the first kind. See A008310.
Row 6 of A142992. (End)
PROG
(PARI) {a(n) = sum(k=0, 6, binomial(12, 2*k)*binomial(n+k, 6))}
(PARI) Vec((1 + 6*x + x^2)*(1 + 60*x + 134*x^2 + 60*x^3 + x^4) / (1 - x)^7 + O(x^40)) \\ Colin Barker, Jun 09 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Seiichi Manyama, Jun 09 2018
STATUS
approved