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%I #34 Sep 08 2022 08:44:33
%S 1,-6,-6,-20,-90,-468,-2652,-15912,-99450,-640900,-4229940,-28455960,
%T -194449060,-1346185800,-9423300600,-66591324240,-474463185210,
%U -3404971093860,-24591457900100,-178611641590200,-1303864983608460,-9561676546462040,-70408709114856840
%N a(n) = (2^n/n!) * Product_{k=0..n-1} (4*k - 3).
%H G. C. Greubel, <a href="/A004983/b004983.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f.: (1 - 8*x)^(3/4).
%F a(n) ~ -(3/4)*Gamma(1/4)^-1*n^(-7/4)*2^(3*n)*(1 + (21/32)*n^-1 + ...).
%F a(n) = (-8)^n/(n*Beta(n, 7/4-n)) if n > 0; a(0)=1. - C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 16 2004
%F a(n) = 8^n*Sum_{k=0..n} ((-1)^k*binomial(k-1/4,k)*binomial(n+3/4,n-k) *binomial(n+k-1,n)). - _Vladimir Kruchinin_, Apr 18 2016
%F a(n) = (-8)^n*Gamma(7/4)/(Gamma(7/4-n)*Gamma(n+1)). - _Ilya Gutkovskiy_, Apr 18 2016
%F a(n) = 8^n * Pochhammer(-3/4, n). - _G. C. Greubel_, Aug 22 2019
%F D-finite with recurrence: n*a(n) +2*(-4*n+7)*a(n-1)=0. - _R. J. Mathar_, Jan 16 2020
%p seq(coeff(convert(series((1-8*x)^(3/4),x,40),polynom),x,i),i=0..25); # C. Ronaldo
%p 1,seq(2^(3*n)*(-1)^n/(n*Beta(n,7/4-n)),n=1..25); # C. Ronaldo
%t Table[2^n/n!*Product[(4*k-3), {k, 0, n-1}], {n, 0, 20}] (* _Vaclav Kotesovec_, Apr 18 2016 *)
%o (Maxima)
%o a(n):=8^n*sum((-1)^k*binomial(k-1/4,k)*binomial(n+3/4,n-k)*binomial(n+k-1,n),k,0,n); /* _Vladimir Kruchinin_, Apr 18 2016 */
%o (PARI) a(n) = 2^n*prod(k=0, n-1, 4*k-3)/n!; \\ _Michel Marcus_, Apr 18 2016
%o (Magma) [1] cat [2^n*&*[4*k-3: k in [0..n-1]]/Factorial(n): n in [1..25]]; // _G. C. Greubel_, Aug 22 2019
%o (Sage) [8^n*rising_factorial(-3/4, n)/factorial(n) for n in (0..25)] # _G. C. Greubel_, Aug 22 2019
%o (GAP) List([0..25], n-> 2^n*Product([0..n-1], k-> 4*k-3)/Factorial(n) ); # _G. C. Greubel_, Aug 22 2019
%K sign,easy
%O 0,2
%A Joe Keane (jgk(AT)jgk.org)
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