OFFSET
0,3
COMMENTS
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*(d^2/dx^2)g(x) + (6+5*x)*(d/dx)g(x) + 4*g(x) = 0, g(0)=1, (d/dx)g(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 Max Alekseyev.
(End)
REFERENCES
J. Ser, Les Calculs Formels des Séries 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).
H. S. Wall, Analytic Theory of Continued Fractions, Chelsea 1973, p. 356.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..440
Emanuele Munarini, Shifting Property for Riordan, Sheffer and Connection Constants Matrices, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.2.
J. Ser, Les Calculs Formels des Séries de Factorielles, Gauthier-Villars, Paris, 1933 [Local copy].
J. Ser, Les Calculs Formels des Séries de Factorielles (Annotated scans of some selected pages)
FORMULA
From Max Alekseyev, 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)
From Peter Bala, Oct 11 2012: (Start)
Numerators in the sequence of convergents of Stieltjes's continued fraction for A073003, the Euler-Gompertz constant G := int {x = 0..oo} 1/(1+x)*exp(-x) dx:
G = 1/(2 - 1^2/(4 - 2^2/(6 - 3^2/(8 - ...)))). See [Wall, Chapter 18, (92.7) with a = 1]. The sequence of convergents to the continued fraction begins [1/2, 4/7, 20/34, 124/209, ...]. The denominators are in A002720.
(End)
G.f.: x = Sum_{n>=1} a(n) * x^n * (1 - (n+1)*x)^2. - Paul D. Hanna, Feb 06 2013
a(n) ~ G * exp(2*sqrt(n) - n - 1/2) * n^(n+1/4) / sqrt(2) * (1 + 31/(48*sqrt(n))), where G = 0.596347362323194... is the Gompertz constant (see A073003). - Vaclav Kotesovec, Oct 19 2013
MATHEMATICA
Flatten[{0, RecurrenceTable[{(-1+n)^2 a[-2+n]-2 n a[-1+n]+a[n]==0, a[1]==1, a[2]==4}, a, {n, 20}]}] (* Vaclav Kotesovec, Oct 19 2013 *)
nxt[{n_, a_, b_}]:={n+1, b, 2(n+1)b-n^2 a}; NestList[nxt, {1, 0, 1}, 30][[All, 2]] (* Harvey P. Dale, Sep 06 2022 *)
PROG
(PARI) A058006(n) = sum(k=0, n, (-1)^k*k! );
a(n) = if (n<=1, n, sum(k=1, n, (k+1) * A058006(k-1) * binomial(n, k) * (n-1)! / (k-1)! ) ); /* Joerg Arndt, Oct 12 2012 */
(PARI) {a(n)=if(n==1, 1, polcoeff(1-sum(m=1, n-1, a(m)*x^m*(1-(m+1)*x+x*O(x^n))^2), n))} \\ Paul D. Hanna, Feb 06 2013
(Magma) I:=[1, 4]; [0] cat [n le 2 select I[n] else 2*n*Self(n-1) - (n-1)^2*Self(n-2): n in [1..30]]; // G. C. Greubel, May 16 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
Edited by Max Alekseyev, Jul 13 2010
STATUS
approved