A032109 "BIJ" (reversible, indistinct, labeled) transform of 1,1,1,1,...

1, 1, 2, 7, 38, 271, 2342, 23647, 272918, 3543631, 51123782, 811316287, 14045783798, 263429174191, 5320671485222, 115141595488927, 2657827340990678, 65185383514567951, 1692767331628422662, 46400793659664205567, 1338843898122192101558, 40562412499252036940911

Offset: 0

Author

Christian G. Bower

Keywords

• nonn

Comments

Starting (1, 2, 7, 38, 271, ...) = A008292 * [1, 1, 2, 4, 8, ...]. - Gary W. Adamson, Dec 25 2008

The inverse binomial transform is 1, 0, 1, 3, 19, 135, 1171, 11823, 136459, ..., see A091346. - R. J. Mathar, Oct 17 2012

Stirling transform of A001710. - Anton Zakharov, Aug 06 2016

For n even (resp. n odd), a(n) counts the ordered partitions of {1,2,...,n} with an even (resp. odd) number of blocks, and a(n)-1 counts the ordered partitions of {1,2,...,n} with an odd (resp. even) number of blocks. - Jose A. Rodriguez, Feb 04 2021

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..424 (first 101 terms from R. J. Mathar)
• C. G. Bower, Transforms (2)

Crossrefs

A000629, A000670, A002050, A032109, A052856, A076726 are all more-or-less the same sequence. - N. J. A. Sloane, Jul 04 2012

A052856(n)=2*a(n), if n>0.

Cf. A008292, A001710.

Programs

Maple

A032109 := proc(n)
    (A000670(n)+1)/2 ;
end proc: # R. J. Mathar, Oct 17 2012
a := n -> (polylog(-n, 1/2)+`if`(n=0,3,2))/4:
seq(round(evalf(a(n), 32)), n=0..18); # Peter Luschny, Nov 03 2015
# alternative Maple program:
b:= proc(n, m) option remember; `if`(n=0, m!,
      add(b(n-1, max(m, j)), j=1..m+1))
    end:
a:= n-> (b(n,0)+1)/2:
seq(a(n), n=0..23);  # Alois P. Heinz, Sep 29 2017

Mathematica

Table[(PolyLog[-n, 1/2] + 2 + KroneckerDelta[n])/4, {n, 0, 20}] (* Vladimir Reshetnikov, Nov 02 2015 *)

PARI

a(n)=if(n<0,0,n!*polcoeff(subst((1-y^2/2)/(1-y),y,exp(x+x*O(x^n))-1),n))

PARI

list(n)=my(v=Vec(subst((1-y^2/2)/(1-y),y,exp(x+x*O(x^n))-1)));vector(n+1,i,v[i]*(i-1)!) \\ Charles R Greathouse IV, Oct 17 2012

Formula

E.g.f.: (e^(2*x)-2*e^x-1)/(2*e^x-4).

a(n) = (A000670(n)+1)/2. - Vladeta Jovovic, Apr 13 2003

a(n) = A052875(n)/2 + 1. - Max Alekseyev, Jan 31 2021

a(n) ~ sqrt(Pi/2)*n^(n+1/2)/(2*log(2)^(n+1)*exp(n)). - Ilya Gutkovskiy, Aug 06 2016

a(n) = Sum_{s in S_n^even} Product_{i=1..n} binomial(i,s(i)-1), where s ranges over the set S_n^even of even permutations of [n]. - Jose A. Rodriguez, Feb 02 2021