A096965 - OEIS

%I #27 Dec 01 2021 10:54:55

%S 1,1,1,7,37,241,2101,18271,201097,2270017,29668681,410815351,

%T 6238931821,101560835377,1765092183037,32838929702671,644215775792401,

%U 13441862819232001,293976795292186897,6788407001443004647,163735077313046119861,4142654439686285737201

%N Number of sets of even number of even lists, cf. A000262.

%H Vincenzo Librandi, <a href="/A096965/b096965.txt">Table of n, a(n) for n = 0..200</a>

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

%F a(n) = (n!*sum(m=floor((n+1)/2)..n, (binomial(n-1,2*m-n-1))/(2*m-n)!)). - _Vladimir Kruchinin_, Mar 10 2013

%F Recurrence: (n-2)*a(n) = (2*n-3)*a(n-1) + (n-1)*(2*n^2 - 8*n + 7)*a(n-2) + (n-2)*(n-1)*(2*n-5)*a(n-3) - (n-4)*(n-3)*(n-2)^2*(n-1)*a(n-4). - _Vaclav Kotesovec_, Sep 29 2013

%F a(n) ~ exp(2*sqrt(n)-n-1/2)*n^(n-1/4)/(2*sqrt(2)) * (1-5/(48*sqrt(n))). - _Vaclav Kotesovec_, Sep 29 2013

%F From _Alois P. Heinz_, Dec 01 2021: (Start)

%F a(n) = A000262(n) - A096939(n).

%F a(n) = |Sum_{k=0..n} (-1)^k * A349776(n,k)|. (End)

%p a:= proc(n) option remember;  `if`(n<4, [1$3, 7][n+1], ((2*n-3)

%p       *a(n-1)+(n-1)*(2*n^2-8*n+7)*a(n-2) + (n-2)*(n-1)*(2*n-5)

%p       *a(n-3)-(n-4)*(n-3)*(n-2)^2*(n-1)*a(n-4))/(n-2))

%p     end:

%p seq(a(n), n=0..25);  # _Alois P. Heinz_, Dec 01 2021

%t Drop[ Range[0, 20]! CoefficientList[ Series[ Exp[(x/(1 - x^2))]Cosh[x^2/(1 - x^2)], {x, 0, 20}], x], 1] (* _Robert G. Wilson v_, Aug 19 2004 *)

%Y Cf. A000262, A088026, A088009, A096939, A349776.

%K easy,nonn

%O 0,4

%A _Vladeta Jovovic_, Aug 18 2004

%E More terms from _Robert G. Wilson v_, Aug 19 2004

%E a(0)=1 prepended by _Alois P. Heinz_, Dec 01 2021