OFFSET
0,2
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Germain Kreweras, Sur une classe de problèmes de dénombrement liés au treillis des partitions des entiers, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, 6 (1965), circa p. 82.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).
FORMULA
a(n) = (n*(1 + n)^2*(2 + n)*(1 + 2*n)*(3 + 2*n))/90.
a(n) = 2*A006858(n).
a(n) = (-1)^(n+1)*A132339(3, n).
G.f.: 2*(1+x)*(1 + 6*x + x^2)/(1-x)^7.
From G. C. Greubel, Dec 14 2021: (Start)
E.g.f.: (1/90)*x*(180 + 1080*x + 1350*x^2 + 555*x^3 + 84*x^4 + 4*x^5)*exp(x).
a(n) = binomial(n+2, 3)*binomial(2*n+3, 3)/5. (End)
From Amiram Eldar, Jul 10 2023: (Start)
Sum_{n>=1} 1/a(n) = 15*Pi^2 - 295/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = -15*Pi^2/2 + 120*Pi - 605/2. (End)
MAPLE
A006332:=-2*(1+z)*(z**2+6*z+1)/(z-1)**7; # conjectured by Simon Plouffe in his 1992 dissertation
MATHEMATICA
Table[(n(1+n)^2(2+n)(1+2n)(3+2n))/90, {n, 0, 30}] (* or *)
{0}~Join~CoefficientList[Series[2(x+1)(x^2 +6x +1)/(1-x)^7, {x, 0, 29}], x] (* Michael De Vlieger, Mar 26 2016 *)
PROG
(PARI) my(x='x+O('x^99)); concat(0, Vec(2*(x+1)*(x^2+6*x+1)/(1-x)^7)) \\ Altug Alkan, Mar 26 2016
(Magma) [Binomial(n+2, 3)*Binomial(2*n+3, 3)/5: n in [0..30]]; // G. C. Greubel, Dec 14 2021
(Sage) [binomial(n+2, 3)*binomial(2*n+3, 3)/5 for n in (0..30)] # G. C. Greubel, Dec 14 2021
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved