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a(n) = n!*Sum_{m=0..floor((n-1)/2)} 1/(n-m)/binomial(n-m-1,m).
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%I #26 Nov 19 2022 20:17:45

%S 1,1,5,10,74,216,2316,8688,128880,581760,11406240,59667840,1482693120,

%T 8782905600,266800262400,1762116249600,63536485017600,462613126348800,

%U 19342202181120000,153884245616640000,7325057766297600000

%N a(n) = n!*Sum_{m=0..floor((n-1)/2)} 1/(n-m)/binomial(n-m-1,m).

%F E.g.f.: log((x-1)^2*(x+1))/(x^2-x-1).

%F a(n) = n!*Sum_{i=1..n} (F(i)/(n-i+1))*(2*(-1)^(i+1)+(-1)^n), F(n) - Fibonacci numbers.

%o (Maxima)

%o a(n):=n!*sum(1/(n-m)/(binomial(n-m-1,m)),m,0,floor((n-1)/2));

%o a(n):=n!*sum((fib(i))/(n-i+1)*(2*(-1)^(i+1)+(-1)^(n)),i,1,n);

%Y Cf. A000045, A358446

%K nonn

%O 1,3

%A _Vladimir Kruchinin_, Nov 19 2022