OFFSET
0,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
V. Kotesovec, Asymptotic of a sums of powers of binomial coefficients * x^k, 2012
FORMULA
General recurrecnce for Sum_{k=0..n} binomial(n,k)^3*x^k (this is case x=6): (n+3)^2*(3*n+4)*a(n+3) -(9*n^3+57*n^2+116*n+74)*(x+1)*a(n+2) +(3*n+5)*(3*n^2*(x^2-7*x+1)+11*n*(x^2-7*x+1)+9*x^2-66*x+9)*a(n+1) -(n+1)^2*(3*n+7)*(x+1)^3*a(n) = 0.
a(n) ~ (1+6^(1/3))^2/(2*2^(1/3)*3^(5/6)*Pi) * (1+6^(1/3))^(3*n)/n. - Vaclav Kotesovec, Sep 19 2012
G.f.: hypergeom([1/3, 2/3],[1],6*27*x^2/(1-7*x)^3)/(1-7*x). - Mark van Hoeij, May 02 2013
a(n) = hypergeom([-n,-n,-n],[1,1], -6). - Peter Luschny, Sep 23 2014
MATHEMATICA
Table[Sum[Binomial[n, k]^3*6^k, {k, 0, n}], {n, 0, 25}]
PROG
(Sage)
A216698 = lambda n: hypergeometric([-n, -n, -n], [1, 1], -6)
[Integer(A216698(n).n(100)) for n in (0..18)] # Peter Luschny, Sep 23 2014
CROSSREFS
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, Sep 15 2012
EXTENSIONS
Minor edits by Vaclav Kotesovec, Mar 31 2014
STATUS
approved