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A216698 a(n) = Sum_{k=0..n} binomial(n,k)^3 * 6^k.

%I A216698 #30 Jul 11 2020 12:38:59
%S A216698 1,7,85,1351,23281,422527,7951069,153458935,3018043777,60225528727,
%T A216698 1215821974885,24777776573095,508935634491025,10522995625652335,
%U A216698 218814097786515085,4572338217781407031,95953172529722919937,2021236451413828339495,42719661851354642952181
%N A216698 a(n) = Sum_{k=0..n} binomial(n,k)^3 * 6^k.
%H A216698 Vincenzo Librandi, <a href="/A216698/b216698.txt">Table of n, a(n) for n = 0..200</a>
%H A216698 V. Kotesovec, <a href="http://www.kotesovec.cz/math_articles/kotesovec_binomial_asymptotics.pdf">Asymptotic of a sums of powers of binomial coefficients * x^k</a>, 2012
%F A216698 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.
%F A216698 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
%F A216698 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
%F A216698 a(n) = hypergeom([-n,-n,-n],[1,1], -6). - _Peter Luschny_, Sep 23 2014
%t A216698 Table[Sum[Binomial[n, k]^3*6^k, {k, 0, n}], {n, 0, 25}]
%o A216698 (Sage)
%o A216698 A216698 = lambda n: hypergeometric([-n,-n,-n], [1,1], -6)
%o A216698 [Integer(A216698(n).n(100)) for n in (0..18)] # _Peter Luschny_, Sep 23 2014
%Y A216698 Cf. A000172 (x=1), A206178 (x=2), A206180 (x=3), A216483 (x=4), A216636 (x=5), A216696.
%K A216698 nonn
%O A216698 0,2
%A A216698 _Vaclav Kotesovec_, Sep 15 2012
%E A216698 Minor edits by _Vaclav Kotesovec_, Mar 31 2014