This site is supported by donations to The OEIS Foundation.

Annual Appeal: Please make a donation (tax deductible in USA) to keep the OEIS running. Over 5000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A153229 a(0) = 0, a(1) = 1, and for n>=2, a(n) = (n-1) * a(n-2) + (n-2) * a(n-1). 9
 0, 1, 0, 2, 4, 20, 100, 620, 4420, 35900, 326980, 3301820, 36614980, 442386620, 5784634180, 81393657020, 1226280710980, 19696509177020, 335990918918980, 6066382786809020, 115578717622022980, 2317323290554617020, 48773618881154822980, 1075227108896452857020 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Previous name was: Weighted Fibonacci numbers. From Peter Bala, Aug 18 2013: (Start) 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) ((-1)^(n+1))*a(n) = p(n,-1), where the polynomials p are defined at A248664.   - Clark Kimberling, Oct 11 2014 LINKS Alois P. Heinz, Table of n, a(n) for n = 0..200 FORMULA a(0) = 0, a(1) = 1, and for n>=2, a(n) = (n-1) * a(n-2) + (n-2) * a(n-1). For n>=1, a(n) = A058006(n-1) * (-1)^(n-1). 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 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 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 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 EXAMPLE a(20) = 19 * a(18) + 18 * a(19) = 19 * 335990918918980 + 18 * 6066382786809020 = 6383827459460620 + 109194890162562360 = 115578717622022980 MAPLE 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 # second Maple program: a:= proc(n) a(n):= `if`(n<2, n, (n-1)*a(n-2) +(n-2)*a(n-1)) end: seq(a(n), n=0..25); # Alois P. Heinz, May 24 2013 MATHEMATICA Join[{a = 0}, Table[b = n! - a; a = b, {n, 0, 100}]] (* Vladimir Joseph Stephan Orlovsky, Jun 28 2011 *) PROG (C) unsigned long a(unsigned int n) { if (n == 0) return 0; if (n == 1) return 1; return (n - 1) * a(n - 2) + (n - 2) * a(n - 1); } (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 CROSSREFS Cf. A000045, A000587, A058006. Sequence in context: A158094 A108879 A058006 * A013329 A102087 A052573 Adjacent sequences:  A153226 A153227 A153228 * A153230 A153231 A153232 KEYWORD nonn AUTHOR Shaojun Ying (dolphinysj(AT)gmail.com), Dec 21 2008 EXTENSIONS Edited by Max Alekseyev, Jul 05 2010 Better name by Joerg Arndt, Aug 17 2013 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.