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%I M3567 N1446 #65 Oct 18 2022 19:12:52
%S 0,1,4,20,124,920,7940,78040,859580,10477880,139931620,2030707640,
%T 31805257340,534514790680,9591325648580,182974870484120,
%U 3697147584561340,78861451031150840,1770536585183202980,41729280102868841080,1030007496863617367420,26568602827124392999640
%N a(n) = 2n*a(n-1) - (n-1)^2*a(n-2).
%C From _Wolfdieter Lang_, Dec 12 2011: (Start)
%C 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.
%C 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).
%C 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.
%C This e.g.f. g(x) is equivalent to the recurrence
%C b(n)= -2*(n+1)*b(n-1) - n^2*b(n-2), b(-1)=0, b(0)=1.
%C 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_.
%C (End)
%D J. Ser, Les Calculs Formels des Séries de Factorielles. Gauthier-Villars, Paris, 1933, p. 78.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D H. S. Wall, Analytic Theory of Continued Fractions, Chelsea 1973, p. 356.
%H G. C. Greubel, <a href="/A002793/b002793.txt">Table of n, a(n) for n = 0..440</a>
%H Emanuele Munarini, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Munarini/muna4.html">Shifting Property for Riordan, Sheffer and Connection Constants Matrices</a>, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.2.
%H J. Ser, <a href="/A002720/a002720_4.pdf">Les Calculs Formels des Séries de Factorielles</a>, Gauthier-Villars, Paris, 1933 [Local copy].
%H J. Ser, <a href="/A002720/a002720.pdf">Les Calculs Formels des Séries de Factorielles</a> (Annotated scans of some selected pages)
%F From _Max Alekseyev_, Jul 06 2010: (Start)
%F For n > 1, a(n) = Sum_{k=1..n} (k+1) * A058006(k-1) * binomial(n,k) * (n-1)! / (k-1)!.
%F E.g.f.: (Gamma(0,1) - Gamma(0,1/(1-x))) * exp(1/(1-x)) / (1-x). (End)
%F From _Peter Bala_, Oct 11 2012: (Start)
%F 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:
%F 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.
%F (End)
%F G.f.: x = Sum_{n>=1} a(n) * x^n * (1 - (n+1)*x)^2. - _Paul D. Hanna_, Feb 06 2013
%F 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
%t 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 *)
%t 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 *)
%o (PARI) A058006(n) = sum(k=0,n, (-1)^k*k! );
%o 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 */
%o (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
%o (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
%Y Bisection of A056952. A199577 (alternating row sums, unsigned).
%Y Cf. A002720, A073003.
%K nonn
%O 0,3
%A _N. J. A. Sloane_, _Robert G. Wilson v_
%E Edited by _Max Alekseyev_, Jul 13 2010
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