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A052657 Expansion of e.g.f. x^2/((1-x)^2*(1+x)).

%I A052657 #37 Jan 22 2023 08:37:54
%S A052657 0,0,2,6,48,240,2160,15120,161280,1451520,18144000,199584000,
%T A052657 2874009600,37362124800,610248038400,9153720576000,167382319104000,
%U A052657 2845499424768000,57621363351552000,1094805903679488000,24329020081766400000,510909421717094400000,12364008005553684480000
%N A052657 Expansion of e.g.f. x^2/((1-x)^2*(1+x)).
%C A052657 Stirling transform of -(-1)^n*a(n-1) = [0, 0, 2, -6, 48, -240, ...] is A052841(n-1) = [0, 0, 2, 6, 38, 270, ...]. - _Michael Somos_, Mar 04 2004
%H A052657 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=604">Encyclopedia of Combinatorial Structures 604</a>.
%F A052657 a(0)=0, a(1)=0, a(2)=2, n*a(n+2) = (n+2)*a(n+1) + (n^3 + 4*n^2 + 5*n + 2)*a(n).
%F A052657 a(n) = (2*n-1+(-1)^n)*n!/4 = n!*floor(n/2) = n!*A004526(n).
%F A052657 E.g.f.: x^2/((1-x)*(1-x^2)).
%F A052657 Sum_{n>=2} 1/a(n) = 4*CoshIntegral(1) - 4*gamma - 2*sinh(1) + 2 = 4*A099284 - 4*A001620 - 2*A073742 + 2. - _Amiram Eldar_, Jan 22 2023
%p A052657 spec := [S,{S=Prod(Z,Z,Sequence(Z),Sequence(Prod(Z,Z)))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%t A052657 a[n_] := Floor[n/2] * n!; Array[a, 25, 0] (* _Amiram Eldar_, Jan 22 2023 *)
%o A052657 (PARI) a(n)=if(n<0,0,n!*polcoeff(x^2/(1-x)/(1-x^2)+x*O(x^n),n))
%o A052657 (PARI) a(n)=n!*(n\2); \\ _Joerg Arndt_, Jan 22 2023
%Y A052657 Cf. A004526, A052841.
%Y A052657 Cf. A001620, A073742, A099284.
%K A052657 easy,nonn
%O A052657 0,3
%A A052657 encyclopedia(AT)pommard.inria.fr, Jan 25 2000