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A004984 a(n) = (2^n/n!)*Product_{k=0..n-1} (4*k - 1).

%I A004984 #54 Dec 29 2024 14:08:29
%S A004984 1,-2,-6,-28,-154,-924,-5852,-38456,-259578,-1788204,-12517428,
%T A004984 -88759944,-636112932,-4599585816,-33511268088,-245749299312,
%U A004984 -1812401082426,-13433090375628,-100001895018564,-747382583822952,-5605369378672140,-42173731515723720
%N A004984 a(n) = (2^n/n!)*Product_{k=0..n-1} (4*k - 1).
%H A004984 Seiichi Manyama, <a href="/A004984/b004984.txt">Table of n, a(n) for n = 0..1000</a>
%F A004984 G.f.: (1 - 8*x)^(1/4).
%F A004984 a(n) ~ -1/4*Gamma(3/4)^-1*n^(-5/4)*2^(3*n)*{1 + 5/32*n^-1 + ...}
%F A004984 a(n) = -1/n*Sum_{k=1..n-1} 2^n*binomial(k+n-1, n-1) * ( Sum_{j=0..k} binomial(k,j)*binomial(j,n-1-3*k+2*j)*(3/2)^(3*k-n+1-j)*(-1)^(n-1-3*k)*(1/4)^(k-j) ), n>1. - _Vladimir Kruchinin_, Sep 14 2010
%F A004984 a(n) = 8^n*Pochhammer(-1/4, n)/n! = -(1/4)*8^n*Gamma(n-1/4)/(Gamma(3/4)*n!). - _Robert Israel_, Sep 29 2014
%F A004984 D-finite with recurrence: n*a(n) +2*(-4*n+5)*a(n-1)=0. - _R. J. Mathar_, Jan 16 2020
%p A004984 seq(-(1/4)*8^n*GAMMA(n-1/4)/(GAMMA(3/4)*n!), n=0..30); # _Robert Israel_, Sep 29 2014
%t A004984 Table[8^n*Pochhammer[-1/4, n]/n!, {n,0,30}] (* _G. C. Greubel_, Aug 22 2019 *)
%t A004984 CoefficientList[Series[Surd[1-8x,4],{x,0,30}],x] (* _Harvey P. Dale_, Dec 08 2019 *)
%o A004984 (PARI) for(n=0,28,print1(2^n/n!*prod(k=0,n-1,(4*k-1)),","))
%o A004984 (Maxima) a(n):=-1/n*(if n=1 then (2)^n else sum(sum(binomial(k,j)* binomial(j,n-1-3*k+2*j)*(3/2)^(3*k-n+1-j)*(-1)^(n-1-3*k)*(1/4)^(k-j),j,0,k)*binomial(k+n-1,n-1)*(2)^n,k,1,n-1)); /* _Vladimir Kruchinin_, Sep 14 2010 */
%o A004984 (Maxima) a(n):=binomial(1/4,n)*(-8)^n; /* _Tani Akinari_, Sep 28 2014 */
%o A004984 (GAP) List([0..25],n->(2^n/Factorial(n))*Product([0..n-1],k->4*k-1)); # _Muniru A Asiru_, Apr 28 2018
%o A004984 (Magma) [1] cat [2^n*(&*[4*k-1: k in [0..n-1]])/Factorial(n): n in [1..30]]; // _G. C. Greubel_, Aug 22 2019
%o A004984 (Sage) [8^n*rising_factorial(-1/4, n)/factorial(n) for n in (0..30)] # _G. C. Greubel_, Aug 22 2019
%Y A004984 Cf. A004981, A025749, A034256, A048779, A303538.
%K A004984 sign,easy
%O A004984 0,2
%A A004984 Joe Keane (jgk(AT)jgk.org)
%E A004984 More terms from _Jason Earls_, Dec 04 2001