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a(1)=1; thereafter, a(n+1) = Sum_{d divides n} (n!/(d!*(n/d)!))*2^d*a(n/d).
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%I #13 Dec 10 2017 18:49:55

%S 1,2,8,24,112,256,3456,7040,121856,889344,9521152,19044352,1485742080,

%T 2971492352,127715852288,2347710726144,26083162292224,52166324715520,

%U 8207231011651584,16414462023827456,3388603178995417088,55218988565243363328,646700039922421596160,1293400079844851580928

%N a(1)=1; thereafter, a(n+1) = Sum_{d divides n} (n!/(d!*(n/d)!))*2^d*a(n/d).

%C Suggested by Eq. (80) of (Maia and Mendez, 2008).

%H Seiichi Manyama, <a href="/A295583/b295583.txt">Table of n, a(n) for n = 1..452</a>

%H M. Maia and M. Mendez, <a href="https://doi.org/10.1016/j.disc.2007.09.062">On the arithmetic product of combinatorial species</a>, Discr. Math., 308 (2008), 5407-5427.

%p with(numtheory);

%p B:=(n,d)->n!/(d!*(n/d)!);

%p f:=proc(n) local d; global B; option remember;

%p if n=1 then 1

%p else add( B(n-1,d)*2^d*f((n-1)/d), d in divisors(n-1)); fi;

%p end;

%p [seq(f(n),n=1..40)];

%Y Cf. A295577, A295578.

%K nonn

%O 1,2

%A _N. J. A. Sloane_, Dec 09 2017