A002793 a(n) = 2n*a(n-1) - (n-1)^2*a(n-2). (Formerly M3567 N1446)

0, 1, 4, 20, 124, 920, 7940, 78040, 859580, 10477880, 139931620, 2030707640, 31805257340, 534514790680, 9591325648580, 182974870484120, 3697147584561340, 78861451031150840, 1770536585183202980, 41729280102868841080, 1030007496863617367420, 26568602827124392999640

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

Bisection of A056952. A199577 (alternating row sums, unsigned).

Cf. A002720, A073003.

Sequence in context: A067116 A347339 A067121 * A162509 A371524 A297924

Adjacent sequences: A002790 A002791 A002792 * A002794 A002795 A002796

Keyword

nonn

Author

N. J. A. Sloane, Robert G. Wilson v

Extensions

Edited by Max Alekseyev, Jul 13 2010

Status

approved