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

0, 1, 0, 2, 4, 20, 100, 620, 4420, 35900, 326980, 3301820, 36614980, 442386620, 5784634180, 81393657020, 1226280710980, 19696509177020, 335990918918980, 6066382786809020, 115578717622022980, 2317323290554617020, 48773618881154822980, 1075227108896452857020

Offset: 0

Author

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

Keywords

• nonn
• easy

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) := Integral_{u >= 0} exp(-u)*u^n/(1 + u) du.

The result is I(n) = A153229(n) + (-1)^n*I(0), where I(0) = Integral_{u >= 0} exp(-u)/(1 + u) du = 0.5963473623... is known as Gompertz's constant. See A073003.

Note also that I(n) = n!*Integral_{u >= 0} 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

Examples

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A000045, A000255, A000587, A058006.

First differences of A136580.

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

Programs

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); }

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 *)
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 *)

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

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

E.g.f.: exp(1)/exp(x)*(Ei(1, 1-x)-Ei(1, 1)). - Alois P. Heinz, Jul 05 2018

a(n) = Sum_{k = 0..n-1} (-1)^(n-k-1) * k!. - Peter Bala, Dec 05 2024

Extensions

Edited by Max Alekseyev, Jul 05 2010

Better name by Joerg Arndt, Aug 17 2013