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%I #20 Sep 08 2022 08:46:23
%S 0,1,0,-3,-14,-51,-168,-521,-1542,-4365,-11740,-29439,-65670,-112273,
%T -28344,1018689,6961550,34606929,151831044,623095683,2453975622,
%U 9402575805,35339538912,130994480547,480676041954,1750847208621,6343667488692,22899720430251,82466180250590
%N Expansion of x*(1 - 4*x)^(3/2)/(3*x - 1)^2.
%H G. C. Greubel, <a href="/A320826/b320826.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = c(n)*h(n) where c(n) = Catalan(n)*(3*n*(n + 1))/(2*(2*n-5)*(2*n-3)*(2*n-1)) = (-4)^(n-1)*binomial(3/2, n-1) and h(n) = hypergeom([2, 1 - n], [7/2 - n], 3/4).
%F A320826(n) = A320825(n) - A320827(n).
%p c := n -> (-4)^(n-1)*binomial(3/2, n-1):
%p h := n -> hypergeom([2, 1 - n], [7/2 - n], 3/4):
%p A320826 := n -> c(n)*h(n): seq(simplify(A320826(n)), n=0..28);
%t CoefficientList[Series[(x (1 - 4 x)^(3/2))/(3 x - 1)^2, {x, 0, 28}], x]
%o (PARI) x='x+O('x^30); concat([0], Vec(x*(1-4*x)^(3/2)/(1-3*x)^2)) \\ _G. C. Greubel_, Oct 27 2018
%o (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); [0] cat Coefficients(R!(x*(1-4*x)^(3/2)/(1-3*x)^2)); // _G. C. Greubel_, Oct 27 2018
%Y Cf. A002421, A320825, A320827.
%K sign
%O 0,4
%A _Peter Luschny_, Oct 22 2018
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