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A328000 a(n) = Sum_{k=0..n}(k!*(n - k)!)/(floor(k/2)!*floor((n - k)/2)!)^2.

%I A328000 #37 Sep 08 2022 08:46:24
%S A328000 1,2,5,16,28,96,160,512,896,2560,4864,12288,25600,57344,131072,262144,
%T A328000 655360,1179648,3211264,5242880,15466496,23068672,73400320,100663296,
%U A328000 343932928,436207616,1593835520,1879048192,7314866176,8053063680,33285996544,34359738368
%N A328000 a(n) = Sum_{k=0..n}(k!*(n - k)!)/(floor(k/2)!*floor((n - k)/2)!)^2.
%H A328000 Colin Barker, <a href="/A328000/b328000.txt">Table of n, a(n) for n = 0..1000</a>
%H A328000 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,12,0,-48,0,64).
%F A328000 a(n) = Sum_{k=0..n} s(k)*s(n-k) where s(n) = A056040(n).
%F A328000 a(n) = [x^n] (4*x^2 - x - 1)^2 / (1 - 4*x^2)^3.
%F A328000 a(n) = 2^(n - 5)*(n*(n + 2) + 32) if n even else 2^(n - 1)*(n + 1).
%F A328000 a(2*n) = A327999(n).
%F A328000 a(2*n-1) = A002699(n), (with a(-1) = 0).
%F A328000 a(2^n-1) = 2^(2^n - 2 + n) for n >= 1.
%F A328000 2*a(2*n)/2^n = A081908(n+1).
%F A328000 4*a(2*n)/4^n = A145018(n+1).
%F A328000 2*a(2*n-1)/4^n = A001477(n).
%F A328000 From _Stefano Spezia_, Oct 19 2019: (Start)
%F A328000 a(n) = n! [x^n] (1/32)*exp(-2*x)*(8 + exp(4*x)*(8 + x)*(3 + 2*x) + x*(13 + 2*x)).
%F A328000 a(n) = 12*a(n-2) - 48*a(n-4) + 64*a(n-6) for n > 5. (End)
%p A328000 swing := n -> n!/iquo(n,2)!^2: a := n -> add(swing(k)*swing(n-k), k=0..n):
%p A328000 seq(`if`(irem(n, 2) = 0, 2 + n*(n + 2)/16, n + 1)*2^(n - 1), n=0..31);
%t A328000 A328000List[len_] := CoefficientList[Series[(4 x^2 - x - 1)^2 / (1 - 4 x^2)^3 , {x, 0, len}], x]; A328000List[31]
%t A328000 LinearRecurrence[{0,12,0,-48,0,64},{1,2,5,16,28,96},40] (* _Harvey P. Dale_, Jun 19 2022 *)
%o A328000 (PARI) x='x + O('x^32);
%o A328000 Vec(serlaplace(((3*x + 8)*sinh(2*x) + (2*x^2 + 16*(x + 1))*cosh(2*x))/16))
%o A328000 (PARI) Vec((1 + x - 4*x^2)^2 / ((1 - 2*x)^3*(1 + 2*x)^3) + O(x^30)) \\ _Colin Barker_, Feb 05 2020
%o A328000 (Magma) [IsOdd(n) select 2^(n - 1)*(n + 1) else 2^(n - 5)*(n*(n + 2) + 32):n in [0..30]]; // _Marius A. Burtea_, Feb 05 2020
%Y A328000 Cf. A327999, A328001.
%Y A328000 Cf. A056040, A002699, A001477, A081908, A145018.
%K A328000 nonn,easy
%O A328000 0,2
%A A328000 _Peter Luschny_, Oct 01 2019