%I #35 Nov 17 2024 07:16:57

%S 1,2,9,46,256,1510,9283,58848,381963,2525916,16958498,115288674,

%T 792042589,5490312864,38352695246,269719400974,1908059370583,

%U 13568804436340,96942782340802,695513575242284,5008808999633736,36195063931874308,262372258663337954

%N Expansion of (4/(3*x/(1-x))) * sin((1/3)*arcsin(sqrt(27*x/4/(1-x))))^2.

%H G. C. Greubel, <a href="/A270386/b270386.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = Sum_{k=0..n} binomial(n-1,n-k)*(binomial(3*k+1,k+1)/(2*k+1)).

%F G.f.: g(x/(1-x)) where g(x) is the g.f. of A006013.

%F a(n) ~ 31^(n + 1/2) / (sqrt(Pi) * n^(3/2) * 2^(2*n + 2)). - _Vaclav Kotesovec_, Mar 16 2016

%F a(n) = 2*hypergeometric([5/3, 7/3, 1-n], [5/2, 3], -27/4) for n>0. - _Peter Luschny_, Mar 16 2016

%F Conjecture D-finite with recurrence: 2*(n+1)*(2*n+1)*a(n) +(-39*n^2+8*n+5)*a(n-1) +(66*n-37)*(n-2)*a(n-2) -31*(n-2)*(n-3)*a(n-3)=0. - _R. J. Mathar_, Jun 07 2016

%t Table[Sum[(Binomial[n - 1, n - k]*((Binomial[3*k + 1, k + 1])/(2*k + 1))), {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Mar 16 2016, after _Vladimir Kruchinin_ *)

%o (Maxima)

%o a(n):=(sum(binomial(n-1,n-k)*((binomial(3*k+1,k+1))/(2*k+1)),k,0,n));

%o (PARI) a(n) = sum(k=0, n, binomial(n-1,n-k)*(binomial(3*k+1,k+1)/(2*k+1))); \\ _Michel Marcus_, Mar 16 2016

%o (Sage)

%o a = lambda n: simplify(2*hypergeometric([5/3, 7/3, 1-n],[5/2, 3],-27/4)) if n>0 else 1

%o [a(n) for n in range(23)] # _Peter Luschny_, Mar 16 2016

%Y Cf. A006013.

%K nonn,changed

%O 0,2

%A _Vladimir Kruchinin_, Mar 16 2016