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%I #12 May 13 2013 01:50:01
%S 3,18,1080,181440,59875200,32691859200,26676557107200,
%T 30411275102208000,46164315605151744000,90020415430045900800000,
%U 219289731987591814348800000,652606242395073239502028800000
%N a(n+1) = 6*A060544(n)*a(n).
%C Sums of coefficients from (3n+1)th moments of binomial(m,k)*binomial(2m,k): see Maple code below.
%H Eric W. Weisstein, <a href="http://mathworld.wolfram.com/BinomialSums.html">MathWorld: Binomial Sums</a>
%H <a href="/index/Di#divseq">Index to divisibility sequences</a>
%F a(n) = (1/18)*27^n*Gamma(n-1/3)*Gamma(n-2/3)*sqrt(3)/Pi.
%e The evaluation of sum(k=0..n, k^7*binomial(n,k)*binomial(2*n,k)) involves the polynomial 32*n^7+96*n^6-336*n^5-360*n^4+1020*n^3-42*n^2-455*n+63, the sum of the coefficients of which is 18 = a(2).
%p with(PolynomialTools); polyn := proc (q) options operator, arrow; 3^q*Pi*GAMMA(2*n)*(sum(k^q*binomial(n, k)*binomial(2*n, k), k = 0 .. n))/(27^n*sqrt(3)*GAMMA(n-floor((1/3)*q+1/3)+2/3)*GAMMA(n-floor((1/3)*q)+1/3)) end proc; coefl := proc (q) options operator, arrow; CoefficientList(expand(polyn(q)), n) end proc; coe := proc (j, h) options operator, arrow; coefl(j)[h] end proc; seq(sum(coe(3*r+1, k), k = 1 .. 5*r+1), r = 1 .. 8) ;
%o (PARI) print1(a=3);for(n=2,10,print1(", ",a*=27*n*(n-3)+60)) \\ _Charles R Greathouse IV_, Dec 26 2011
%Y Cf. A060544, A064350.
%K nonn,easy
%O 1,1
%A _John M. Campbell_, Dec 26 2011
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