This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A270386 #37 Aug 19 2025 16:32:05
%S A270386 1,2,9,46,256,1510,9283,58848,381963,2525916,16958498,115288674,
%T A270386 792042589,5490312864,38352695246,269719400974,1908059370583,
%U A270386 13568804436340,96942782340802,695513575242284,5008808999633736,36195063931874308,262372258663337954
%N A270386 Expansion of (4/(3*x/(1-x))) * sin((1/3)*arcsin(sqrt(27*x/4/(1-x))))^2.
%H A270386 G. C. Greubel, <a href="/A270386/b270386.txt">Table of n, a(n) for n = 0..1000</a>
%F A270386 a(n) = Sum_{k=0..n} binomial(n-1,n-k)*(binomial(3*k+1,k+1)/(2*k+1)).
%F A270386 G.f.: g(x/(1-x)) where g(x) is the g.f. of A006013.
%F A270386 a(n) ~ 31^(n + 1/2) / (sqrt(Pi) * n^(3/2) * 2^(2*n + 2)). - _Vaclav Kotesovec_, Mar 16 2016
%F A270386 a(n) = 2*hypergeometric([5/3, 7/3, 1-n], [5/2, 3], -27/4) for n>0. - _Peter Luschny_, Mar 16 2016
%F A270386 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 A270386 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 A270386 (Maxima)
%o A270386 a(n):=(sum(binomial(n-1,n-k)*((binomial(3*k+1,k+1))/(2*k+1)),k,0,n));
%o A270386 (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 A270386 (Sage)
%o A270386 a = lambda n: simplify(2*hypergeometric([5/3, 7/3, 1-n],[5/2, 3],-27/4)) if n>0 else 1
%o A270386 [a(n) for n in range(23)] # _Peter Luschny_, Mar 16 2016
%Y A270386 Cf. A006013.
%K A270386 nonn
%O A270386 0,2
%A A270386 _Vladimir Kruchinin_, Mar 16 2016