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A006154 Number of labeled ordered partitions of an n-set into odd parts.
(Formerly M1792)
16

%I M1792

%S 1,1,2,7,32,181,1232,9787,88832,907081,10291712,128445967,1748805632,

%T 25794366781,409725396992,6973071372547,126585529106432,

%U 2441591202059281,49863806091395072,1074927056650469527

%N Number of labeled ordered partitions of an n-set into odd parts.

%D Getu, S.; Shapiro, L. W.; Combinatorial view of the composition of functions. Ars Combin. 10 (1980), 131-145.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Vincenzo Librandi, <a href="/A006154/b006154.txt">Table of n, a(n) for n = 0..200</a>

%F E.g.f.: 1/(1 - sinh(x)).

%F With alternating signs, e.g.f.: 1/(1+sinh(x)). - _Ralf Stephan_, Apr 29 2004

%F a(0) = a(1) = 1, a(n) = Sum_{k=1..ceiling(n/2)} C(n,2*k-1)*a(n-2*k+1). - _Ralf Stephan_, Apr 29 2004

%F a(n) ~ (sqrt(2)/2)*n!/log(1+sqrt(2))^(n+1). - Conjectured by _Simon Plouffe_, Feb 17 2007.

%F From A. N. W. Hone (A.N.W.Hone(AT)kent.ac.uk), Feb 22 2007: (Start)

%F 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)

%F a(n) = Sum_{k=1..n} Sum_{i=0..k} (-1)^i*(k-2*i)^n*binomial(k,i)/2^k, n > 0, a(0)=1. - _Vladimir Kruchinin_, May 28 2011

%F Row sums (apart from a(0)) of A196776. - _Peter Bala_, Oct 06 2011

%F Row sums of A193474. - _Peter Luschny_, Oct 07 2011

%F 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

%F From _Sergei N. Gladkovskii_, Jun 01 2012: (Start)

%F Let E(x) be the e.g.f., then

%F E(x) = -1/x + 1/(x*(1-x))+ x^3/((1-x)*((1-x)*G(0) - x^2)); G(k) = (2*k+2)*(2*k+3)+x^2-(2*k+2)*(2*k+3)*x^2/G(k+1); (continued fraction).

%F E(x) = -1/x + 1/(x*(1-x))+ x^3/((1-x)*((1-x)*G(0) - x^2)); G(k) = 8*k+6+x^2/(1 + (2*k+2)*(2*k+3)/G(k+1)); (continued fraction).

%F E(x) = 1/(1 - x*G(0)); G(k) = 1 + x^2/(2*(2*k+1)*(4*k+3) + 2*x^2*(2*k+1)*(4*k+3)/(-x^2 - 4*(k+1)*(4*k+5)/G(k+1))); (continued fraction).

%F (End).

%F E.g.f. 1/(1 - x*G(0)) where G(k) = 1 - x^2/( (2*k+1)*(2*k+3) - 2*k+1)*(2*k+3)^2/(2*k+3 - (2*k+2)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Oct 01 2012

%p readlib(coeftayl):

%p with(combinat, bell);

%p A:=series(1/(1-sinh(x)),x,20);

%p G(x):=A : f[0]:=G(x): for n from 0 to 21 do f[n]:=coeftayl(G(x), x=0, n);;

%p p[n]:=f[n]*((n)!) od: x:=0:seq(p[n], n=0..20); # _Sergei N. Gladkovskii_, Jun 01 2012

%t 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}] (* _Jean-Fran├žois Alcover_, Dec 07 2011, after _Vladimir Kruchinin_ *)

%t With[{nn=20},CoefficientList[Series[1/(1-Sinh[x]),{x,0,nn}],x]Range[0,nn]!] (* _Harvey P. Dale_, Nov 16 2012 *)

%o (PARI) a(n)=if(n<2,n>=0,sum(k=1,ceil(n/2),binomial(n,2*k-1)*a(n-2*k+1))) \\ _Ralf Stephan_

%o (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 */

%Y Cf. A000045, A000111, A000670, A003724, A193474, A196776.

%K nonn,easy,nice

%O 0,3

%A _Simon Plouffe_

%E More terms from _Christian G. Bower_, Oct 15 1999

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Last modified October 21 11:05 EDT 2019. Contains 328294 sequences. (Running on oeis4.)