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A007840
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Number of factorizations of permutations of n letters into ordered cycles.
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92
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1, 1, 3, 14, 88, 694, 6578, 72792, 920904, 13109088, 207360912, 3608233056, 68495486640, 1408631978064, 31197601660080, 740303842925184, 18738231641600256, 503937595069600896, 14349899305396086912, 431322634732516137216, 13646841876634025159424
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OFFSET
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0,3
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COMMENTS
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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. - Geoffrey Critzer, Mar 18 2009
The terms of this sequence for n >= 1 are the row sums of A008275^2, the unsigned version of A039814. - Peter Bala, Jul 22 2014
Signed sequence is the base for an Appell sequence of polynomials with the e.g.f. e^(x*t)/[log(1+t) + 1] = exp(P(.,x),t) that is the umbral compositional inverse for A238385, reverse of A111492, i.e., umbrally evaluated UP(n,P(.,t))= x^n = P(n,UP(.,t)) where UP(n,t) are the polynomials of A238385. Umbrally evaluated means letting (A(.,t))^n = A(n,t) after substituting A for the independent variable of the polynomial. - Tom Copeland, Nov 15 2014
a(n) is the number of unimodal rooted forests on n labeled nodes (i.e., those forests that avoid the patterns 213 and 312). - Kassie Archer, Aug 30 2018
Number of permutations of [n] where fixed points at index j are j-colored and all other points are unicolored. - Alois P. Heinz, Apr 24 2020
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LINKS
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FORMULA
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a(n) = 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. - Philippe Deléham, Dec 09 2003
a(n) = n!*Sum_{k=0..n-1} a(k)/k!/(n-k) for n >= 1 with a(0)=1. - Paul D. Hanna, Jul 19 2006
E.g.f.: B(A(x)) where B(x) = 1/(1-x) and A(x) = log(1/(1-x)). - Geoffrey Critzer, 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. - Paul D. Hanna, Dec 31 2011
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MAPLE
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a:= proc(n) a(n):= n!*`if`(n=0, 1, add(a(k)/(k!*(n-k)), k=0..n-1)) end:
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MATHEMATICA
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Table[Sum[Abs[StirlingS1[n, k]] k!, {k, 0, n}], {n, 0, 20}] (* Geoffrey Critzer, Mar 18 2009 *)
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PROG
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(PARI) a(n)=n!*polcoeff(1/(1+log(1-x +x*O(x^n))), n) /* Paul D. Hanna, 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, Jul 19 2006 */
(Sage)
f, R, C = 1, [1], [1]+[0]*len
for n in (1..len):
f *= n
for k in range(n, 0, -1):
C[k] = -C[k-1]*((k-1)/k if k>1 else 1)
C[0] = sum((-1)^k*C[k] for k in (1..n))
R.append(C[0]*f)
return R
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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Extended June 1995
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STATUS
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approved
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