OFFSET
2,1
COMMENTS
Apparently n divides a(n), so a(n)/n = 1, 1, 5, 18, 99, 600, 4318, 35112, 320724, 3245400, 36057526, 436352400, 5713654296, ... - R. J. Mathar, Oct 31 2015
REFERENCES
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999 (Sec. 5.2)
LINKS
Harvey P. Dale, Table of n, a(n) for n = 2..250
Steven Finch, Rounds, Color, Parity, Squares, arXiv:2111.14487 [math.CO], 2021.
P. Flajolet, S. Gerhold and B. Salvy, On the non-holonomic character of logarithms, powers and the n-th prime function, arXiv:math/0501379 [math.CO], 2005.
FORMULA
E.g.f.: -1+1/(1-x)^x.
a(n) ~ n! * (1 - 1/n + (1-log(n)-gamma)/n^2), where gamma is the Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Apr 21 2014
a(n) = b(n), n>0, a(0)=0, where b(n) = (n-1)!*Sum_{i=1..n-1} (1+1/i)*b(n-i-1)/(n-i-1)!, b(0)=1. - Vladimir Kruchinin, Feb 25 2015
E.g.f.: Sum_{n>=1} x^n/n! * Product_{k=0..n-1} (k + x). - Paul D. Hanna, Oct 26 2015
a(n) = n! * Sum_{k=0..floor(n/2)} |Stirling1(n-k,k)|/(n-k)!. - Seiichi Manyama, May 10 2022
EXAMPLE
a(4)=20: 12 ways to make 2 hugs, 8 ways to make a 3-ring.
MATHEMATICA
Drop[With[{nn=20}, CoefficientList[Series[1/(1-x)^x-1, {x, 0, nn}], x] Range[ 0, nn]!], 2] (* Harvey P. Dale, Sep 17 2011 *)
PROG
(PARI) a(n)=if(n<0, 0, n!*polcoeff(-1+1/(1-x+x*O(x^n))^x, n))
(PARI) {a(n) = n!*polcoeff( sum(m=1, n, x^m/m! * prod(k=0, m-1, x + k) +x*O(x^n) ), n)}
for(n=2, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Oct 26 2015
(PARI) a(n) = n!*sum(k=0, n\2, abs(stirling(n-k, k, 1))/(n-k)!); \\ Seiichi Manyama, May 10 2022
(Maxima)
b(n):=if n=0 then 1 else (n-1)!*sum((1+1/i)*b(n-i-1)/(n-i-1)!, i, 1, n-1);
makelist(a(n), n, 2, 10); /* Vladimir Kruchinin, Feb 25 2015 */
(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(-1+1/(1-x)^x)); [Factorial(n+1)*b[n]: n in [1..m-2]]; // G. C. Greubel, Aug 29 2018
CROSSREFS
KEYWORD
nonn,nice,easy
AUTHOR
Len Smiley, Dec 12 2001
STATUS
approved