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

0, 0, 2, 6, 60, 380, 3990, 37002, 450296, 5373720, 76018410, 1096730030, 17814654132, 299645294676, 5511836578430, 105550556136690, 2171244984679920, 46545825736022192, 1059273836225051346, 25100215228045842390, 626204775725372971820, 16239127347086448236460

Offset: 0

Author

Vladeta Jovovic, Feb 19 2005

Keywords

• easy
• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..444
• Index entries for sequences related to Laguerre polynomials

Crossrefs

Cf. A052852, A059570, A066897, A066898, A081358, A092691.

Programs

Magma

l:= func< n,b | Evaluate(LaguerrePolynomial(n), b) >;
[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

Maple

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
# second Maple program:
b:= proc(n) option remember; `if`(n=0, [1, 0], add(
      (p-> p+`if`(j::even, [0, p[1]], 0))(b(n-j)*
        binomial(n-1, j-1)*j!), j=1..n))
    end:
a:= n-> b(n, 0)[2]:
seq(a(n), n=0..25);  # Alois P. Heinz, May 10 2016

Mathematica

Rest[CoefficientList[Series[x^2/(1-x^2)*E^(x/(1-x)), {x, 0, 20}], x]* Range[0, 20]!] (* Vaclav Kotesovec, Sep 29 2013 *)
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 *)

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

Formula

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

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

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

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

Extensions

More terms from Emeric Deutsch, Mar 27 2005

a(0)=0 prepended by Alois P. Heinz, May 10 2016