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A006154 Number of labeled ordered partitions of an n-set into odd parts.

%I A006154 M1792 #96 Jun 23 2025 18:34:16
%S A006154 1,1,2,7,32,181,1232,9787,88832,907081,10291712,128445967,1748805632,
%T A006154 25794366781,409725396992,6973071372547,126585529106432,
%U A006154 2441591202059281,49863806091395072,1074927056650469527,24392086908129247232,581176736647853024581
%N A006154 Number of labeled ordered partitions of an n-set into odd parts.
%C A006154 Conjecture: taking the sequence modulo an integer k gives an eventually periodic sequence. For example, the sequence taken modulo 10 is [1, 1, 2, 7, 2, 1, 2, 7, 2, 1, 2, 7, 2, ...], with an apparent period [1, 2, 7, 2] beginning at a(1), of length 4. Cf. A000670. - _Peter Bala_, Apr 12 2023
%D A006154 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006154 Vincenzo Librandi, <a href="/A006154/b006154.txt">Table of n, a(n) for n = 0..200</a>
%H A006154 Farid Aliniaeifard and Shu Xiao Li, <a href="https://arxiv.org/abs/2506.12868">Peak algebra in noncommuting variables</a>, arXiv:2506.12868 [math.CO], 2025. See pp. 2, 9, 45.
%H A006154 Philippe Flajolet, <a href="https://algo.inria.fr/seminars/sem00-01/flajolet.pdf">Symbolic Enumerative Combinatorics and Complex Asymptotic Analysis</a>, Algorithms Seminar 2000-2001, F. Chyzak (ed.), INRIA, (2002), pp. 161-170.
%H A006154 S. Getu and L. W. Shapiro, <a href="/A006152/a006152.pdf">Combinatorial view of the composition of functions</a>, Ars Combin. 10 (1980), 131-145. (Annotated scanned copy)
%H A006154 Prabha Sivaraman Nair and Rejikumar Karunakaran, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL27/Nair/nair11.html">On k-Fibonacci Brousseau Sums</a>, J. Int. Seq. (2024) Art. No. 24.6.4. See p. 8.
%F A006154 E.g.f.: 1/(1 - sinh(x)).
%F A006154 With alternating signs, e.g.f.: 1/(1+sinh(x)). - _Ralf Stephan_, Apr 29 2004
%F A006154 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 A006154 a(n) ~ (sqrt(2)/2)*n!/log(1+sqrt(2))^(n+1). - Conjectured by _Simon Plouffe_, Feb 17 2007.
%F A006154 From _Andrew Hone_, Feb 22 2007: (Start)
%F A006154 This formula can be proved using the techniques in the article by Philippe Flajolet (see links) [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 A006154 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 A006154 Row sums (apart from a(0)) of A196776. - _Peter Bala_, Oct 06 2011
%F A006154 Row sums of A193474. - _Peter Luschny_, Oct 07 2011
%F A006154 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 A006154 From _Sergei N. Gladkovskii_, Jun 01 2012: (Start)
%F A006154 Let E(x) be the e.g.f., then
%F A006154 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 A006154 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 A006154 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 A006154 (End).
%F A006154 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
%F A006154 O.g.f A(x) satisfies A(x) = 1 + ( A(x/(1-x))/(1-x) - A(x/(1+x))/(1+x) )/2. - _Paul D. Hanna_, Aug 19 2024
%p A006154 readlib(coeftayl):
%p A006154 with(combinat, bell);
%p A006154 A:=series(1/(1-sinh(x)),x,20);
%p A006154 G(x):=A : f[0]:=G(x): for n from 0 to 21 do f[n]:=coeftayl(G(x), x=0, n);;
%p A006154 p[n]:=f[n]*((n)!) od: x:=0:seq(p[n], n=0..20); # _Sergei N. Gladkovskii_, Jun 01 2012
%p A006154 # second Maple program:
%p A006154 a:= proc(n) option remember; `if`(n=0, 1, add((i->
%p A006154       a(n-i)*binomial(n, i))(2*j+1), j=0..(n-1)/2))
%p A006154     end:
%p A006154 seq(a(n), n=0..23);  # _Alois P. Heinz_, Feb 01 2022
%t A006154 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-François Alcover_, Dec 07 2011, after _Vladimir Kruchinin_ *)
%t A006154 With[{nn=20},CoefficientList[Series[1/(1-Sinh[x]),{x,0,nn}],x]Range[0,nn]!] (* _Harvey P. Dale_, Nov 16 2012 *)
%o A006154 (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 A006154 (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 A006154 Cf. A000045, A000111, A000670, A003724, A193474, A196776.
%K A006154 nonn,easy,nice
%O A006154 0,3
%A A006154 _Simon Plouffe_
%E A006154 More terms from _Christian G. Bower_, Oct 15 1999