|
| |
|
|
A006154
|
|
Number of labeled ordered partitions of an n-set into odd parts.
(Formerly M1792)
|
|
13
| |
|
|
1, 1, 2, 7, 32, 181, 1232, 9787, 88832, 907081, 10291712, 128445967, 1748805632, 25794366781, 409725396992, 6973071372547, 126585529106432, 2441591202059281, 49863806091395072, 1074927056650469527
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,3
|
|
|
COMMENTS
| With alternating signs, e.g.f.: 1/(1+sinh(x)). - R. Stephan, Apr 29 2004
|
|
|
REFERENCES
| Getu, S.; Shapiro, L. W.; Combinatorial view of the composition of functions. Ars Combin. 10 (1980), 131-145.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
|
|
|
FORMULA
| E.g.f.: 1/(1 - sinh(x)).
a(0) = a(1) = 1, a(n) = sum(k=1..ceil(n/2), C(n,2*k-1)*a(n-2*k+1) ). - R. Stephan, Apr 29 2004
a(n) ~ (sqrt(2)/2)*n!/log(1+sqrt(2))^(n+1). - Conjectured by Simon Plouffe, Feb 17 2007.
From A. N. W. Hone (A.N.W.Hone(AT)kent.ac.uk), Feb 22 2007: (Start)
This formula can be proved using the techniques in the article by Philippe Flajolet, Symbolic Enumerative Combinatorics and Complex Asymptotic Analysis, Algorithms Seminar 2000-2001, F. Chyzak (ed.), INRIA, (2002), pp. 161-170 [see Theorem 5 and Table 2, noting that 1/(1-sinh(x)) just has a simple pole at x=log(1+sqrt(2)]. (End)
a(n) = sum(k=1..n, sum(i=0..k,(-1)^i*(k-2*i)^n*C(k,i))/2^k), n>0, a(0)=1. [Vladimir Kruchinin, May 28 2011]
Row sums (apart from a(0)) of A196776 - Peter Bala, Oct 06 2011
Row sums of A193474 - Peter Luschny, Oct 07 2011
a(n) = D^n(1/(1-x)) evaluated at x = 0, where D is the operator sqrt(1+x^2)*d/dx. Cf. A003724 and A000111. - Peter Bala, Dec 06 2011
|
|
|
MATHEMATICA
| a[n_] := Sum[ (-1)^i*(k - 2*i)^n*Binomial[k, i]/2^k, {k, 1, n}, {i, 0, k}]; a[0] = 1; Table[a[n], {n, 0, 19}] (* From Jean-François Alcover, Dec 07 2011, after Vladimir Kruchinin *)
|
|
|
PROG
| (PARI) a(n)=if(n<2, n>=0, sum(k=1, ceil(n/2), binomial(n, 2*k-1)*a(n-2*k+1))) (from R. Stephan)
(Maxima) a(n):=sum(sum((-1)^i*(k-2*i)^n*binomial(k, i), i, 0, k)/2^k, k, 1, n); [Vladimir Kruchinin, May 28 2011]
|
|
|
CROSSREFS
| Cf. A000045, A000670, A196776, A193474, A003724, A000111.
Sequence in context: A097900 A198891 A000153 * A000987 A006957 A189780
Adjacent sequences: A006151 A006152 A006153 * A006155 A006156 A006157
|
|
|
KEYWORD
| nonn,easy,nice
|
|
|
AUTHOR
| Simon Plouffe (simon.plouffe(AT)gmail.com)
|
|
|
EXTENSIONS
| More terms from Christian G. Bower (bowerc(AT)usa.net), Oct 15 1999.
|
| |
|
|
