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A204820
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a(n) = -4*a(n-1)*A001505(n-2), with a(1)=8.
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1
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8, -192, 161280, -638668800, 6974263296000, -162193467211776000, 6893871130369327104000, -483949753351926762700800000, 52208499391605859160162304000000, -8200911084433448356878294712320000000
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OFFSET
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1,1
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COMMENTS
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Sums of coefficients from (4n+1)th moments of binomial(m,k) * binomial(3*m,k); see Maple code below.
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LINKS
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FORMULA
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a(n)=-(1/8)*GAMMA(2*n-3/2)*GAMMA(n-1/2)*(-1)^n*64^n/Pi
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EXAMPLE
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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).
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MAPLE
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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);
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CROSSREFS
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KEYWORD
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easy,sign
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AUTHOR
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STATUS
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approved
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