%I #39 Jul 11 2020 12:37:27

%S 1,6,66,936,14346,231876,3885456,66767616,1169068986,20769386796,

%T 373277526876,6772297456656,123834925330416,2279408745325536,

%U 42194656181618496,784905308800229376,14663340953943086106,274968958499402854716,5173516852494573136836

%N a(n) = Sum_{k=0..n} binomial(n,k)^3 * 5^k.

%C Diagonal of rational function 1/(1 + y + z + x*y + y*z + 5*x*z + 6*x*y*z). - _Gheorghe Coserea_, Jul 01 2018

%C Diagonal of rational function 1 / ((1-x)*(1-y)*(1-z) - 5*x*y*z). - _Seiichi Manyama_, Jul 11 2020

%H Vincenzo Librandi, <a href="/A216636/b216636.txt">Table of n, a(n) for n = 0..200</a>

%H Vaclav 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 Recurrence: (n+3)^2*(3*n+4)*a(n+3) = 6*(9*n^3+57*n^2+116*n+74)*a(n+2) + 3*(27*n^3+144*n^2+261*n+160)*a(n+1) + 216*(3*n+7)*(n+1)^2*a(n).

%F Asymptotic: a(n) ~ (1+5^(1/3))^2/(2*sqrt(3)*5^(1/3)*Pi) * (3*5^(2/3)+3*5^(1/3)+6)^n/n. - _Vaclav Kotesovec_, Sep 19 2012

%F G.f.: hypergeom([1/3, 2/3],[1],5*27*x^2/(1-6*x)^3)/(1-6*x). - _Mark van Hoeij_, May 02 2013

%F a(n) = hypergeom([-n,-n,-n],[1,1], -5). - _Peter Luschny_, Sep 23 2014

%F G.f. y=A(x) satisfies: x*(3*x + 1)*(216*x^3 + 27*x^2 + 18*x - 1)*y'' + (1944*x^4 + 1026*x^3 + 135*x^2 + 36*x - 1)*y' + 6*(108*x^3 + 69*x^2 + 2*x + 1)*y. - _Gheorghe Coserea_, Jul 01 2018

%t Table[Sum[Binomial[n,k]^3*5^k,{k,0,n}],{n,0,20}]

%o (Sage)

%o A216636 = lambda n: hypergeometric([-n,-n,-n],[1,1], -5)

%o [Integer(A216636(n).n(100)) for n in (0..18)] # _Peter Luschny_, Sep 23 2014

%o (PARI) a(n) = sum(k=0, n, binomial(n,k)^3 * 5^k); \\ _Gheorghe Coserea_, Jul 01 2018

%Y Cf. A206178, A206180, A216483.

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, Sep 11 2012

%E Minor edits by _Vaclav Kotesovec_, Mar 31 2014