A203778 a(n) = -24*A015219(n-2)*a(n-1), with a(1) = 2.
2, -48, 40320, -159667200, 1743565824000, -40548366802944000, 1723467782592331776000, -120987438337981690675200000, 13052124847901464790040576000000, -2050227771108362089219573678080000000, 449688758403823707201064412255354880000000
Offset: 1
Examples
The evaluation of sum(binomial(n,k)*binomial(3*n,k)*k^8,k=0..n) involves the polynomial 729*n^13+729*n^12-12879*n^11+9801*n^10+50247*n^9-84825*n^8-105*n^7+74167*n^6-36968*n^5-2296*n^4+1472*n^3-120*n^2, the sum of the coefficients of which is a(2)=-48.
Links
- Eric W. Weisstein, MathWorld: Binomial Sums
- Index to divisibility sequences
Programs
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Maple
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*g),a),a=1..(2*(4*g)-floor(((4*g)+3)/4))),g=1..6);seq(simplify(-(1/32)*GAMMA(2*n-3/2)*GAMMA(n-1/2)*(-1)^n*64^n/Pi),n=1..6);
Formula
a(n) = -(1/32)*Gamma(2*n-3/2)*Gamma(n-1/2)*(-1)^n*64^n/Pi.
Comments