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a(0) = 0, a(1) = 1, and for n>=2, a(n) = (n-1) * a(n-2) + (n-2) * a(n-1).
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%I #42 Aug 17 2024 20:45:58

%S 0,1,0,2,4,20,100,620,4420,35900,326980,3301820,36614980,442386620,

%T 5784634180,81393657020,1226280710980,19696509177020,335990918918980,

%U 6066382786809020,115578717622022980,2317323290554617020,48773618881154822980,1075227108896452857020

%N a(0) = 0, a(1) = 1, and for n>=2, a(n) = (n-1) * a(n-2) + (n-2) * a(n-1).

%C Previous name was: Weighted Fibonacci numbers.

%C From _Peter Bala_, Aug 18 2013: (Start)

%C The sequence occurs in the evaluation of the integral I(n) := int {u = 0..inf} exp(-u)*u^n/(1 + u) du. The result is I(n) = A153229(n) + (-1)^n*I(0), where I(0) = int {0..inf} exp(-u)/(1 + u) du = 0.5963473623... is known as Gompertz's constant. See A073003. Note also that I(n) = n!*int {u = 0..inf} exp(-u)/(1 + u)^(n+1) du. (End)

%C ((-1)^(n+1))*a(n) = p(n,-1), where the polynomials p are defined at A248664. - _Clark Kimberling_, Oct 11 2014

%H Alois P. Heinz, <a href="/A153229/b153229.txt">Table of n, a(n) for n = 0..200</a>

%F a(0) = 0, a(1) = 1, and for n>=2, a(n) = (n-1) * a(n-2) + (n-2) * a(n-1).

%F For n>=1, a(n) = A058006(n-1) * (-1)^(n-1).

%F G.f.: G(0)*x/(1+x)/2, where G(k)= 1 + 1/(1 - x*(k+1)/(x*(k+1) + 1/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 24 2013

%F G.f.: 2*x/(1+x)/G(0), where G(k)= 1 + 1/(1 - 1/(1 - 1/(2*x*(k+1)) + 1/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 29 2013

%F G.f.: W(0)*x/(1+sqrt(x))/(1+x), where W(k) = 1 + sqrt(x)/( 1 - sqrt(x)*(k+1)/(sqrt(x)*(k+1) + 1/W(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Aug 17 2013

%F a(n) ~ (n-1)! * (1 - 1/n + 1/n^3 + 1/n^4 - 2/n^5 - 9/n^6 - 9/n^7 + 50/n^8 + 267/n^9 + 413/n^10), where numerators are Rao Uppuluri-Carpenter numbers, see A000587. - _Vaclav Kotesovec_, Mar 16 2015

%F E.g.f.: exp(1)/exp(x)*(Ei(1, 1-x)-Ei(1, 1)). - _Alois P. Heinz_, Jul 05 2018

%e a(20) = 19 * a(18) + 18 * a(19) = 19 * 335990918918980 + 18 * 6066382786809020 = 6383827459460620 + 109194890162562360 = 115578717622022980

%p t1 := sum(n!*x^n, n=0..100): F := series(t1/(1+x), x, 100): for i from 0 to 40 do printf(`%d, `, i!-coeff(F, x, i)) od: # _Zerinvary Lajos_, Mar 22 2009

%p # second Maple program:

%p a:= proc(n) a(n):= `if`(n<2, n, (n-1)*a(n-2) +(n-2)*a(n-1)) end:

%p seq(a(n), n=0..25); # _Alois P. Heinz_, May 24 2013

%t Join[{a = 0}, Table[b = n! - a; a = b, {n, 0, 100}]] (* _Vladimir Joseph Stephan Orlovsky_, Jun 28 2011 *)

%t RecurrenceTable[{a[0]==0,a[1]==1,a[n]==(n-1)a[n-2]+(n-2)a[n-1]},a,{n,30}] (* _Harvey P. Dale_, May 01 2020 *)

%o (C) unsigned long a(unsigned int n) {

%o if (n == 0) return 0;

%o if (n == 1) return 1;

%o return (n - 1) * a(n - 2) + (n - 2) * a(n - 1); }

%o (PARI) a(n)=if(n,my(t=(-1)^n);-t-sum(i=1,n-1,t*=-i),0); \\ _Charles R Greathouse IV_, Jun 28 2011

%Y Cf. A000045, A000255, A000587, A058006.

%Y First differences of A136580.

%Y Column k=0 of A303697 (for n>0).

%K nonn

%O 0,4

%A Shaojun Ying (dolphinysj(AT)gmail.com), Dec 21 2008

%E Edited by _Max Alekseyev_, Jul 05 2010

%E Better name by _Joerg Arndt_, Aug 17 2013