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

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..444

Index entries for sequences related to Laguerre polynomials

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

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 *)

Prog

(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
(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

Crossrefs

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

Sequence in context: A379725 A355286 A308532 * A025540 A334224 A226959

Adjacent sequences: A102287 A102288 A102289 * A102291 A102292 A102293

Keyword

easy,nonn,changed

Author

Vladeta Jovovic, Feb 19 2005

Extensions

More terms from Emeric Deutsch, Mar 27 2005

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

Status

approved