|
%I #22 Jan 30 2020 21:29:16
%S 1,5,21,83,319,1209,4551,17085,64125,240995,907741,3428655,12990121,
%T 49370963,188229489,719805987,2760498351,10615101273,40920439119,
%U 158106581157,612166272291,2374756691313,9228369037659,35918537840577
%N a(n) = 3^n*Sum_{ k=0..n } binomial(2*k,k)/3^k.
%C Simpler definition from _N. J. A. Sloane_, Jan 21 2009. Colin Mallows and I studied this sequence on Feb 21 1981 in connection with integration over a regular (solid) hexagon.
%C Hankel transform is A137717. - _Paul Barry_, Apr 26 2009
%H Vincenzo Librandi, <a href="/A132310/b132310.txt">Table of n, a(n) for n = 0..300</a>
%F a(n) = C(2n,n) * sum_{k=0..2n} trinomial(n,k)/C(2n,k) where trinomial(n,k) = [x^k] (1 + x + x^2)^n, where [x^k] denotes "coefficient of x^k in ...".
%F G.f.: A(x) = 1/sqrt(1 - 10*x + 33*x^2 - 36*x^3).
%F a(n) = sum_{k=0..2n} trinomial(n,k) * k!*(2*n-k)! / (n!)^2.
%F 2*a(n) = sum(A182411(n+1,i), i=0..n). - _Bruno Berselli_, May 02 2012
%F D-finite with recurrence: n*a(n) = (7*n-2)*a(n-1) - 6*(2*n-1)*a(n-2) . - _Vaclav Kotesovec_, Oct 20 2012
%F a(n) ~ 4^(n+1)/sqrt(Pi*n) . - _Vaclav Kotesovec_, Oct 20 2012
%e a(1) = C(2,1)*(1/1 + 1/2 + 1/1) = 2*(5/2) = 5;
%e a(2) = C(4,2)*(1/1 + 2/4 + 3/6 + 2/4 + 1/1) = 6*(7/2) = 21;
%e a(3) = C(6,3)*(1/1 + 3/6 + 6/15 + 7/20 + 6/15 + 3/6 + 1/1) = 20*(83/20) = 83.
%e 2*a(6) = sum(A182411(7,i), i=0..6) = 3432+858+572+572+728+1092+1848 = 9102 = 2*4551. - _Bruno Berselli_, May 02 2012
%t CoefficientList[Series[1/Sqrt[1-10*x+33*x^2-36*x^3], {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 20 2012 *)
%o (PARI) a(n)=binomial(2*n,n)*sum(k=0,2*n, polcoeff((1+x+x^2)^n,k)/binomial(2*n,k) )
%o (PARI) a(n)=sum(k=0,2*n,polcoeff((1+x+x^2)^n,k) * k!*(2*n-k)! / (n!)^2 )
%K nonn
%O 0,2
%A _Paul D. Hanna_, Aug 18 2007
|
