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A102290 Total number of even lists in all sets of lists, cf. A000262.

%I A102290 #19 Jan 08 2025 10:12:16
%S A102290 0,0,2,6,60,380,3990,37002,450296,5373720,76018410,1096730030,
%T A102290 17814654132,299645294676,5511836578430,105550556136690,
%U A102290 2171244984679920,46545825736022192,1059273836225051346,25100215228045842390,626204775725372971820,16239127347086448236460
%N A102290 Total number of even lists in all sets of lists, cf. A000262.
%H A102290 Alois P. Heinz, <a href="/A102290/b102290.txt">Table of n, a(n) for n = 0..444</a>
%H A102290 <a href="/index/La#Laguerre">Index entries for sequences related to Laguerre polynomials</a>
%F A102290 E.g.f.: x^2/(1-x^2)*exp(x/(1-x)).
%F A102290 Recurrence: (n-2)*a(n) = (n-2)*n*a(n-1) + (n-1)^2*n*a(n-2) - (n-3)*(n-2)*(n-1)*n*a(n-3). - _Vaclav Kotesovec_, Sep 29 2013
%F A102290 a(n) ~ sqrt(2)/4 * n^(n+1/4)*exp(2*sqrt(n)-n-1/2) * (1 - 41/(48*sqrt(n))). - _Vaclav Kotesovec_, Sep 29 2013
%F A102290 a(n) = n! * Sum_{j=0..n-2} (-1)^(n+j)*LaguerreL(j, -1) for n>1 with a(0)=a(1)=0. - _G. C. Greubel_, Mar 09 2021
%p A102290 Gser:=series(x^2*exp(x/(1-x))/(1-x^2),x=0,22):seq(n!*coeff(Gser,x^n),n=1..21); # _Emeric Deutsch_
%p A102290 # second Maple program:
%p A102290 b:= proc(n) option remember; `if`(n=0, [1, 0], add(
%p A102290       (p-> p+`if`(j::even, [0, p[1]], 0))(b(n-j)*
%p A102290         binomial(n-1, j-1)*j!), j=1..n))
%p A102290     end:
%p A102290 a:= n-> b(n, 0)[2]:
%p A102290 seq(a(n), n=0..25);  # _Alois P. Heinz_, May 10 2016
%t A102290 Rest[CoefficientList[Series[x^2/(1-x^2)*E^(x/(1-x)), {x, 0, 20}], x]* Range[0, 20]!] (* _Vaclav Kotesovec_, Sep 29 2013 *)
%t A102290 Table[If[n<2, 0, n!*Sum[(-1)^(n-j)*LaguerreL[j, -1], {j,0,n-2}]], {n,0,30}] (* _G. C. Greubel_, Mar 09 2021 *)
%o A102290 (Sage) [0,0]+[factorial(n)*sum((-1)^(n+j)*gen_laguerre(j,0,-1) for j in (0..n-2)) for n in (2..30)] # _G. C. Greubel_, Mar 09 2021
%o A102290 (Magma)
%o A102290 l:= func< n,b | Evaluate(LaguerrePolynomial(n), b) >;
%o A102290 [0,0]cat[Factorial(n)*(&+[(-1)^(n+j)*l(j,-1): j in [0..n-2]]): n in [2..30]]; // _G. C. Greubel_, Mar 09 2021
%Y A102290 Cf. A052852, A059570, A066897, A066898, A081358, A092691.
%K A102290 easy,nonn
%O A102290 0,3
%A A102290 _Vladeta Jovovic_, Feb 19 2005
%E A102290 More terms from _Emeric Deutsch_, Mar 27 2005
%E A102290 a(0)=0 prepended by _Alois P. Heinz_, May 10 2016