OFFSET
1,1
COMMENTS
See Maple code below for formula for such polynomials.
LINKS
Eric W. Weisstein, MathWorld: Binomial Sums
Wikipedia, Hypergeometric Distribution
FORMULA
a(n) = 120*A151989(n-2)*a(n-1), with a(1)=24.
a(n) = 12*5^(5*n-5)*GAMMA(n-4/5)*GAMMA(n-3/5)*GAMMA(n-2/5)*GAMMA(n-1/5)*(cos((2/5)*Pi)+cos((1/5)*Pi))/Pi^2.
EXAMPLE
The evaluation of sum(binomial(n, k)*binomial(4*n, k)*k^5, k = 0 .. n) involves the polynomial 256*n^5-640*n^3+400*n^2+108*n-100, the sum of the coefficients of which is 24 = a(1).
MAPLE
with(PolynomialTools); polyn:=w->simplify(Pi^2*sum(binomial(n, k)*binomial(4*n, k)*k^w, k=0..n)*5^w/3125^n*csc((1/5)*Pi)*csc((2/5)*Pi)*GAMMA(4*n)/GAMMA(n-(floor((w+1)/5)*5-2)/5)/GAMMA(n-(floor(w/5)*5-1)/5)/GAMMA(n-(floor((w+2)/5)*5-3)/5)/GAMMA(n-(floor((w+3)/5)*5-4)/5)); coefl:=d->CoefficientList(expand(polyn(d)), n); seq(sum(coefl(5*h)[m], m=1..nops(coefl(5*h))), h=1..5); seq(simplify(12*5^(5*n-5)*GAMMA(n-4/5)*GAMMA(n-3/5)*GAMMA(n-2/5)*GAMMA(n-1/5)*(cos((2/5)*Pi)+cos((1/5)*Pi))/Pi^2), n=1..5);
CROSSREFS
KEYWORD
nonn
AUTHOR
John M. Campbell, Feb 09 2012
STATUS
approved