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A216483 a(n) = Sum_{k=0..n} binomial(n,k)^3 * 4^k.

%I A216483 #46 Apr 24 2025 04:07:00
%S A216483 1,5,49,605,8065,113525,1656145,24774125,377601025,5839329125,
%T A216483 91349718769,1442580779645,22959923825281,367847984671445,
%U A216483 5926784048373265,95960317086368525,1560335109283897345,25466972987548413125,417048643127042376625,6850021673230814868125
%N A216483 a(n) = Sum_{k=0..n} binomial(n,k)^3 * 4^k.
%C A216483 Diagonal of rational function 1/(1 + y + z + x*y + y*z + 4*x*z + 5*x*y*z). - _Gheorghe Coserea_, Jul 01 2018
%C A216483 Diagonal of rational function 1 / ((1-x)*(1-y)*(1-z) - 4*x*y*z). - _Seiichi Manyama_, Jul 11 2020
%H A216483 Vincenzo Librandi, <a href="/A216483/b216483.txt">Table of n, a(n) for n = 0..200</a>
%H A216483 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 A216483 Recurrence: (n+3)^2*(3*n+4)*a(n+3) = 5*(9*n^3+57*n^2+116*n+74)*a(n+2) + (99*n^3+528*n^2+938*n+555)*a(n+1) + 125*(3*n+7)*(n+1)^2*a(n).
%F A216483 a(n) ~ (1 + 2^(2/3))^(3*n+2) / (2^(5/3)*sqrt(3)*Pi*n). - _Vaclav Kotesovec_, Sep 19 2012, simplified Apr 24 2025
%F A216483 G.f.: hypergeom([1/3, 2/3],[1],108*x^2/(1-5*x)^3)/(1-5*x). - _Mark van Hoeij_, May 02 2013
%F A216483 a(n) = hypergeom([-n,-n,-n],[1,1],-4). - _Peter Luschny_, Sep 23 2014
%F A216483 G.f. y=A(x) satisfies: 0 = x*(5*x + 2)*(125*x^3 + 33*x^2 + 15*x - 1)*y'' + (1875*x^4 + 1330*x^3 + 273*x^2 + 60*x - 2)*y' + (625*x^3 + 495*x^2 + 42*x + 10)*y. - _Gheorghe Coserea_, Jul 01 2018
%t A216483 Table[Sum[Binomial[n,k]^3*4^k,{k,0,n}],{n,0,20}]
%o A216483 (Sage)
%o A216483 A216483 = lambda n: hypergeometric([-n,-n,-n], [1,1], -4)
%o A216483 [Integer(A216483(n).n(100)) for n in (0..19)] # _Peter Luschny_, Sep 23 2014
%o A216483 (PARI) a(n) = sum(k=0, n, binomial(n,k)^3 * 4^k); \\ _Gheorghe Coserea_, Jul 01 2018
%Y A216483 Cf. A206178, A206180, A216636.
%K A216483 nonn
%O A216483 0,2
%A A216483 _Vaclav Kotesovec_, Sep 11 2012
%E A216483 Minor edits by _Vaclav Kotesovec_, Mar 31 2014