A106539 a(1)=1, a(2)=1, a(n) = (n-1)*a(n-1) - (n-2)*a(n-2) - ... - a(1) for n>=3.

1, 1, 1, 0, -6, -36, -192, -1104, -7248, -54816, -472512, -4573824, -49064448, -577130496, -7381281792, -101940854784, -1511556077568, -23945902043136, -403579232182272, -7209532170092544, -136064164749017088, -2705030337674674176, -56501002847058788352

Offset

1,5

Comments

Beginning with a(0)=0, a(1)=1 gives 0, 1, 2, 4, 8, 16, ..., 2^(n-1).

Links

G. C. Greubel, Table of n, a(n) for n = 1..400

Formula

D-finite with recurrence: a(n) = n*a(n-1) - 2*(n-2)*a(n-2), a(1)=1, a(2)=1. - Georg Fischer, Jun 18 2021

a(n) = e^(-2)*( - Gamma(n)*E_{n}(-2) + 2^(n-1)*(-Ei(2) + e^2 - Pi*i), where Ei(x) and E_{n}(x) are exponential integrals. - G. C. Greubel, Sep 03 2021

Example

a(7) = 6*(-36) - 5(-6) - 4*0 - 3*1 - 2*1 - 1*1 = -216 + 30 - 0 - 3 - 2 - 1 = -192.

Maple

nmax:=24; a[1]:=1: a[2]:=1: for n from 3 to nmax do a[n]:=(n-1)*a[n-1]-add(k*a[k], k=1..n-2) od: seq(a[n], n=1..nmax); # Emeric Deutsch, Feb 03 2006

Mathematica

RecurrenceTable[{a[n]==n*a[n-1] - 2*(n-2)*a[n-2], a[1]==a[2]==1}, a[n], {n, 1, 20}] (* Georg Fischer, Jun 18 2021 *)

Prog

(Magma) [n le 2 select 1 else n*Self(n-1) - 2*(n-2)*Self(n-2): n in [1..30]]; // G. C. Greubel, Sep 03 2021
(Sage)
def a(n): return 1 if (n<3) else n*a(n-1) - 2*(n-2)*a(n-2)
[a(n) for n in (1..30)] # G. C. Greubel, Sep 03 2021

Crossrefs

Cf. A001571.

Sequence in context: A209904 A146883 A159721 * A215453 A267229 A048980

Adjacent sequences: A106536 A106537 A106538 * A106540 A106541 A106542

Keyword

easy,sign

Author

Alexandre Wajnberg, May 08 2005

Extensions

More terms from Emeric Deutsch, Feb 03 2006

Definition adapted to offset by Georg Fischer, Jun 18 2021

Status

approved