A204820 - OEIS

%I #6 Mar 31 2012 10:26:21

%S 8,-192,161280,-638668800,6974263296000,-162193467211776000,

%T 6893871130369327104000,-483949753351926762700800000,

%U 52208499391605859160162304000000,-8200911084433448356878294712320000000

%N a(n) = -4*a(n-1)*A001505(n-2), with a(1)=8.

%C Sums of coefficients from (4n+1)th moments of binomial(m,k) * binomial(3*m,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)=-(1/8)*GAMMA(2*n-3/2)*GAMMA(n-1/2)*(-1)^n*64^n/Pi

%e The evaluation of sum(binomial(n, k)*binomial(3*n, k)*k^9, k=0..n) involves the polynomial 2187*n^11+6561*n^10-45927*n^9-28431*n^8+322947*n^7-257985*n^6-473445*n^5+726003*n^4-110482*n^3-189924*n^2+52624*n-4320, the sum of the coefficients of which is -192 = a(2).

%p with(PolynomialTools); polyn:=q->expand(simplify((1/(GAMMA(n-((2*floor((q+1)/4)-1))/(2))))*(1/sqrt(3))*GAMMA(n+1/3)*GAMMA(n+2/3)*(1/3)*(1/(27^(-n)))*GAMMA(n)*1/64^n*sum(binomial(n, k)*binomial(3*n, k)*k^q, k=0..n)*(1/(GAMMA(2*n-((2*floor(q/2)-1)/(2)))))*(2^((floor((1/2)*q+1/2)-1)+q)))); coefl:=h->CoefficientList(expand(polyn(h)), n); coe:=(d, b)->coefl(d)[b];seq(sum(coe((4*d+1),b),b=1..(4*d+1)+floor(((4*d+1)+1)/4)+floor((4*d+1)/2)),d=1..6);seq(-(1/8)*GAMMA(2*n-3/2)*GAMMA(n-1/2)*(-1)^n*64^n/Pi,n=1..6);

%Y Cf. A203778, A015219, A202948, A202946.

%K easy,sign

%O 1,1

%A _John M. Campbell_, Jan 19 2012