A173184 Partial sums of A000166.

1, 1, 2, 4, 13, 57, 322, 2176, 17009, 150505, 1485466, 16170036, 192384877, 2483177809, 34554278858, 515620794592, 8212685046337, 139062777326001, 2494364438359954, 47245095998005060, 942259727190907181, 19737566982241851721, 433234326593362631602

Offset

0,3

Comments

Partial sums of subfactorial or rencontres numbers, or derangements (number of permutations of n elements with no fixed points). The subsequence of primes begins: 2, 13, 192384877.

Links

Table of n, a(n) for n=0..22.

Formula

G.f.: 1/U(0)/(1-x) where U(k) = 1 + x - x*(k+1)/(1 - x*(k+1)/U(k+1)); (continued fraction). - Sergei N. Gladkovskii, Oct 15 2012

G.f.: 1/(1 - x^2) + (1/(1 - x))*Sum_{k>=1} k^k*x^k/(1 + (k + 1)*x)^(k+1). - Ilya Gutkovskiy, Apr 13 2017

From Emanuele Munarini, Oct 06 2017: (Start)

E.g.f.: exp(-t)/(1-t) - exp(t-2)*(coshIntegral(2-2*t) + sinhIntegral(2-2*t) - expIntegralEi(2)).

a(n+2) - (n+3)*a(n+1) + (n+2)*a(n) = (-1)^n. (End)

D-finite with recurrence a(n+3) - (n+3)*a(n+2) + (n+2)*a(n) = 0. - Emanuele Munarini, Aug 24 2017

Example

a(3) = 1 + 0 + 1 + 2 = 4.

Mathematica

a[0] = 1; a[n_] := a[n] = n*a[n - 1] + (-1)^n; Accumulate@ Array[a, 21, 0] (* Robert G. Wilson v, Apr 01 2011 *)
dr[{n_, a1_, a2_}]:={n+1, a2, n(a1+a2)}; Accumulate[Transpose[NestList[dr, {0, 0, 1}, 30]][[3]]] (* Harvey P. Dale, Jul 17 2014 *)
Table[Sum[Subfactorial[k], {k, 0, n}], {n, 0, 100}] (* Emanuele Munarini, Aug 24 2017 *)

Prog

(Maxima)
s[0]:1$
s[n]:=n*s[n-1]+(-1)^n$
makelist(sum(s[k], k, 0, n), n, 0, 12); /* Emanuele Munarini, Aug 24 2017 */

Crossrefs

Cf. A000166, A001277.

Sequence in context: A360508 A328438 A039831 * A111977 A201691 A020120

Adjacent sequences: A173181 A173182 A173183 * A173185 A173186 A173187

Keyword

nonn

Author

Jonathan Vos Post, Feb 12 2010

Status

approved