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A006154 Number of labeled ordered partitions of an n-set into odd parts.
(Formerly M1792)
15
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; text; internal format)
OFFSET

0,3

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).

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

FORMULA

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

With alternating signs, e.g.f.: 1/(1+sinh(x)). - Ralf Stephan, Apr 29 2004

a(0) = a(1) = 1, a(n) = sum(k=1..ceil(n/2), C(n,2*k-1)*a(n-2*k+1) ). - Ralf 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*binomial(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

From Sergei N. Gladkovskii, Jun 01 2012: (Start)

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

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).

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).

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).

(End).

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

MAPLE

readlib(coeftayl):

with(combinat, bell);

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

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

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

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}] (* Jean-Fran├žois Alcover, Dec 07 2011, after Vladimir Kruchinin *)

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

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))) \\ Ralf 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, A000111, A000670, A003724, A193474, A196776.

Sequence in context: A097900 A198891 A000153 * A000987 A006957 A189780

Adjacent sequences:  A006151 A006152 A006153 * A006155 A006156 A006157

KEYWORD

nonn,easy,nice

AUTHOR

Simon Plouffe

EXTENSIONS

More terms from Christian G. Bower, Oct 15 1999

STATUS

approved

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Last modified October 19 19:55 EDT 2018. Contains 316378 sequences. (Running on oeis4.)