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A153229 a(0) = 0, a(1) = 1, and for n>=2, a(n) = (n-1) * a(n-2) + (n-2) * a(n-1). 6
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

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, 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

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Last modified December 18 03:29 EST 2014. Contains 252079 sequences.