|
| |
|
|
A002793
|
|
a(n) = 2n*a(n-1) - (n-1)^2*a(n-2).
(Formerly M3567 N1446)
|
|
3
| |
|
|
0, 1, 4, 20, 124, 920, 7940, 78040, 859580, 10477880, 139931620, 2030707640, 31805257340, 534514790680, 9591325648580, 182974870484120, 3697147584561340, 78861451031150840
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,3
|
|
|
COMMENTS
| Contribution from Wolfdieter Lang, Dec 12 2011 (Start)
r(n) = a(n+1)*(-1)^n, n>=0, gives the alternating row sums of the coefficient triangle A199577, i.e., r(n)=La_n(1;0,-1), with the monic first associated Laguerre polynomials with parameter alpha=0 evaluated at x=-1.
The e.g.f. for these row sums r(n) is g(x) = -(2+x)*exp(1/(1+x))*(Ei(1,1/(1+x))-Ei(1,1))/(1+x)^3 + 1/(1+x)^2, with the exponential integral Ei(1,x) = GAMMA(0,x).
This e.g.f. satisfies the homogeneous ordinary second order differential equation (1+x)^2*diff(g(x),x$2)+(6+5*x)*diff(g(x),x)+4*g(x) = 0, g(0)=1, diff(g(x),x)|_{x=0}=-4.
This e.g.f. g(x) is equivalent to the recurrence
b(n)= -2*(n+1)*b(n-1) - n^2*b(n-2), b(-1)=0, b(0)=1.
Therefore, the e.g.f. of a(n) is A(x)=int(g(-x),x), with A(0)=0. This agrees with the e.g.f. given below in the formula section by M. Alekseyev.
(End)
|
|
|
REFERENCES
| J. Ser, Les Calculs Formels des S\'{e}ries de Factorielles. Gauthier-Villars, Paris, 1933, p. 78.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
|
|
|
FORMULA
| Contribution from Max Alekseyev (maxale(AT)gmail.com), Jul 06 2010: (Start)
For n>1, a(n) = \sum_{k=1}^n (k+1) * A058006(k-1) * binomial(n,k) * (n-1)! / (k-1)!
E.g.f.: (GAMMA(0,1) - GAMMA(0,1/(1-x))) * exp(1/(1-x)) / (1-x) (End)
|
|
|
CROSSREFS
| Bisection of A056952. A199577 (alternating row sums, unsigned).
Sequence in context: A121553 A067116 A067121 * A162509 A151341 A135886
Adjacent sequences: A002790 A002791 A002792 * A002794 A002795 A002796
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Robert G. Wilson v (rgwv(AT)rgwv.com)
|
|
|
EXTENSIONS
| Edited by Max Alekseyev (maxale(AT)gmail.com), Jul 13 2010
|
| |
|
|
