|
| |
|
|
A007840
|
|
Number of factorizations of permutations of n letters into ordered cycles.
|
|
9
| |
|
|
1, 1, 3, 14, 88, 694, 6578, 72792, 920904, 13109088, 207360912, 3608233056, 68495486640, 1408631978064, 31197601660080, 740303842925184, 18738231641600256, 503937595069600896, 14349899305396086912
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,3
|
|
|
COMMENTS
| a(n) is the number of ways to seat n people at an unspecified number of circular tables and then linearly order the nonempty tables. [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Mar 18 2009]
|
|
|
REFERENCES
| Knopfmacher, A.; Ridley, J. N.; Reciprocal sums over partitions and compositions. SIAM J. Discrete Math. 6 (1993), no. 3, 388-399.
|
|
|
LINKS
| P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 119
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 122
|
|
|
FORMULA
| Sum_{k=1}^{n} k!s(n, k), s(n, k) = unsigned Stirling number of first kind; E.g.f. 1/{1+log(1-z)}
For n>0, a(n) is the permanent of the n X n matrix with entries a(i, i) = i and a(i, j) = 1 elsewhere. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Dec 09 2003
a(n) = A052860(n)/n for n>=1.
a(n) = n!*Sum_{k=0..n-1} a(k)/k!/(n-k) for n>=1 with a(0)=1. - Paul D. Hanna (pauldhanna(AT)juno.com), Jul 19 2006
E.g.f. is B(A(x)) where B(x)=1/(1-x) and A(x)=log[1/(1-x)] [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Mar 18 2009]
a(n) = D^n(1/(1-x)) evaluated at x = 0, where D is the operator exp(x)*d/dx. Cf. A006252. - Peter Bala, Nov 25 2011
E.g.f.: 1/(1+log(1-x)) = 1/(1 - x/(1 - x/(2 - x/(3 - 4*x/(4 - 4*x/(5 - 9*x/(6 - 9*x/(7 - 16*x/(8 - 16*x/(9 - ...)))))))))), a continued fraction. [From Paul D. Hanna, Dec 31 2011]
|
|
|
MATHEMATICA
| Table[Sum[Abs[StirlingS1[n, k]] k!, {k, 0, n}], {n, 0, 20}] [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Mar 18 2009]
|
|
|
PROG
| (PARI) a(n)=n!*polcoeff(1/(1+log(1-x +x*O(x^n))), n) - Paul D. Hanna (pauldhanna(AT)juno.com), Jul 19 2006
(PARI) {a(n)=local(CF=1+x*O(x)); for(k=0, n-1, CF=1/((n-k)-((n-k+1)\2)^2*x*CF)); n!*polcoeff(1/(1-x*CF), n)} /* Paul D. Hanna */
|
|
|
CROSSREFS
| Cf. A052860.
Sequence in context: A185323 A199548 A038170 * A007549 A081005 A074518
Adjacent sequences: A007837 A007838 A007839 * A007841 A007842 A007843
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| Arnold Knopfmacher [ ARNOLDK(AT)gauss.cam.wits.ac.za ]
|
|
|
EXTENSIONS
| Extended 6/95.
|
| |
|
|
