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A216483 a(n) = Sum_{k=0..n} binomial(n,k)^3 * 4^k. 6

%I

%S 1,5,49,605,8065,113525,1656145,24774125,377601025,5839329125,

%T 91349718769,1442580779645,22959923825281,367847984671445,

%U 5926784048373265,95960317086368525,1560335109283897345,25466972987548413125,417048643127042376625,6850021673230814868125

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

%C 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 Diagonal of rational function 1 / ((1-x)*(1-y)*(1-z) - 4*x*y*z). - _Seiichi Manyama_, Jul 11 2020

%H Vincenzo Librandi, <a href="/A216483/b216483.txt">Table of n, a(n) for n = 0..200</a>

%H 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 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 a(n) ~ (1+2^(2/3))^2/(2*2^(2/3)*sqrt(3)*Pi) * (3*4^(2/3)+3*4^(1/3)+5)^n/n. - _Vaclav Kotesovec_, Sep 19 2012

%F 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 a(n) = hypergeom([-n,-n,-n],[1,1],-4). - _Peter Luschny_, Sep 23 2014

%F 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 Table[Sum[Binomial[n,k]^3*4^k,{k,0,n}],{n,0,20}]

%o (Sage)

%o A216483 = lambda n: hypergeometric([-n,-n,-n], [1,1], -4)

%o [Integer(A216483(n).n(100)) for n in (0..19)] # _Peter Luschny_, Sep 23 2014

%o (PARI) a(n) = sum(k=0, n, binomial(n,k)^3 * 4^k); \\ _Gheorghe Coserea_, Jul 01 2018

%Y Cf. A206178, A206180, A216636.

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, Sep 11 2012

%E Minor edits by _Vaclav Kotesovec_, Mar 31 2014

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Last modified November 27 06:46 EST 2020. Contains 338678 sequences. (Running on oeis4.)