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A205795
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Sums of coefficients of polynomials from 5n-th moments of X ~ Hypergeometric(4m, 5m, m).
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0
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24, 2880, 43545600, 5230697472000, 2432902008176640000, 3102242008666197196800000, 8841761993739701954543616000000, 49205466506600690141269768273920000000, 485663859076129603777149565235783270400000000, 7911522544013240381082219675638737768808448000000000
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OFFSET
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1,1
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COMMENTS
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See Maple code below for formula for such polynomials.
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LINKS
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FORMULA
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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.
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EXAMPLE
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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).
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MAPLE
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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);
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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