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A328002 a(n) = 2^n * Sum_{k=0..n} Product_{j=1..k} (2/j)^((-1)^j).

%I A328002 #26 Apr 22 2021 16:06:30
%S A328002 1,3,8,22,50,130,280,700,1470,3570,7392,17556,36036,84084,171600,
%T A328002 394680,802230,1823250,3695120,8314020,16812796,37505468,75716368,
%U A328002 167657672,338019500,743642900,1497686400,3276189000,6592494600,14348370600,28851858720,62512360560
%N A328002 a(n) = 2^n * Sum_{k=0..n} Product_{j=1..k} (2/j)^((-1)^j).
%H A328002 Manuel Kauers, <a href="http://www.algebra.uni-linz.ac.at/people/mkauers/publications/kauers15n.pdf">Algorithms for D-finite functions</a>.
%H A328002 Wikipedia, <a href="https://en.wikipedia.org/wiki/Holonomic_function">Holonomic_function</a>
%F A328002 a(n) = sqrt(2/Pi)*Sum_{k=0..n} s(k)*s(n-k-1/2) where s(n) = 2^n*(Gamma(n/2 + 1/2)/Gamma(n/2 + 1))^cos(n*Pi).
%F A328002 a(n) = [x^n] (4*x^2 - x - 1)/((1 - 4*x^2)^(3/2)*(2*x - 1)).
%F A328002 a(n) ~ 2^n * n^(3/2) / (3*sqrt(2*Pi)). - _Vaclav Kotesovec_, Oct 19 2019
%F A328002 D-finite with recurrence: n*a(n) +(n-4)*a(n-1) +2*(-4*n+3)*a(n-2) +4*(-n+2)*a(n-3) +16*(n-2)*a(n-4)=0. - _R. J. Mathar_, Jan 09 2020
%F A328002 D-finite with recurrence: a(n) = (4*n*(n*(n + 5) - 10)*a(n-2) + 2*(n - 6)*(n + 2)*a(n-1))/(n*(n*(n + 3) - 14)). - _Peter Luschny_, Jan 10 2020
%F A328002 Let q(n) = a(n+1)/a(n) and d(n) = numerator(q(n)) - 2*denominator(q(n)).
%F A328002 Conjecture: d(n) is periodic with period 12 and repetition pattern (1, 2, 3, 3, 3, 2, 1, 1, 3, 6, 3, 1). - _Peter Luschny_, Jan 10 2020
%F A328002 a(n) = Sum_{k=0..n} 2^(n-k)*A056040(k). - _Peter Luschny_, Apr 22 2021
%p A328002 s := z -> 2^z*(GAMMA(z/2+1/2)/GAMMA(z/2+1))^cos(z*Pi):
%p A328002 a := n -> sqrt(2/Pi)*add(s(k)*s(n-k-1/2), k=0..n): seq(a(n), n=0..19);
%p A328002 # Alternative:
%p A328002 a := proc(n) option remember; if n=0 then 1 elif n=1 then 3 else
%p A328002 (4*n*(n*(n+5)-10)*a(n-2) + 2*(n-6)*(n+2)*a(n-1))/(n*(n*(n+3)-14)) fi end:
%p A328002 seq(a(n), n=0..31); # _Peter Luschny_, Jan 10 2020
%Y A328002 Cf. A056040, row sums of A338654.
%K A328002 nonn
%O A328002 0,2
%A A328002 _Peter Luschny_, Oct 02 2019
%E A328002 Simpler name by _Peter Luschny_, Apr 22 2021