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%I #73 Mar 31 2021 06:26:23
%S 1,1,2,7,38,271,2342,23647,272918,3543631,51123782,811316287,
%T 14045783798,263429174191,5320671485222,115141595488927,
%U 2657827340990678,65185383514567951,1692767331628422662,46400793659664205567,1338843898122192101558,40562412499252036940911
%N "BIJ" (reversible, indistinct, labeled) transform of 1,1,1,1,...
%C Starting (1, 2, 7, 38, 271, ...) = A008292 * [1, 1, 2, 4, 8, ...]. - _Gary W. Adamson_, Dec 25 2008
%C The inverse binomial transform is 1, 0, 1, 3, 19, 135, 1171, 11823, 136459, ..., see A091346. - _R. J. Mathar_, Oct 17 2012
%C Stirling transform of A001710. - _Anton Zakharov_, Aug 06 2016
%C 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
%H Alois P. Heinz, <a href="/A032109/b032109.txt">Table of n, a(n) for n = 0..424</a> (first 101 terms from R. J. Mathar)
%H C. G. Bower, <a href="/transforms2.html">Transforms (2)</a>
%F E.g.f.: (e^(2*x)-2*e^x-1)/(2*e^x-4).
%F a(n) = (A000670(n)+1)/2. - _Vladeta Jovovic_, Apr 13 2003
%F a(n) = A052875(n)/2 + 1. - _Max Alekseyev_, Jan 31 2021
%F a(n) ~ sqrt(Pi/2)*n^(n+1/2)/(2*log(2)^(n+1)*exp(n)). - _Ilya Gutkovskiy_, Aug 06 2016
%F 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
%p A032109 := proc(n)
%p (A000670(n)+1)/2 ;
%p end proc: # _R. J. Mathar_, Oct 17 2012
%p a := n -> (polylog(-n, 1/2)+`if`(n=0,3,2))/4:
%p seq(round(evalf(a(n), 32)), n=0..18); # _Peter Luschny_, Nov 03 2015
%p # alternative Maple program:
%p b:= proc(n, m) option remember; `if`(n=0, m!,
%p add(b(n-1, max(m, j)), j=1..m+1))
%p end:
%p a:= n-> (b(n,0)+1)/2:
%p seq(a(n), n=0..23); # _Alois P. Heinz_, Sep 29 2017
%t Table[(PolyLog[-n, 1/2] + 2 + KroneckerDelta[n])/4, {n, 0, 20}] (* _Vladimir Reshetnikov_, Nov 02 2015 *)
%o (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))
%o (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
%Y A000629, A000670, A002050, A032109, A052856, A076726 are all more-or-less the same sequence. - _N. J. A. Sloane_, Jul 04 2012
%Y A052856(n)=2*a(n), if n>0.
%Y Cf. A008292, A001710.
%K nonn
%O 0,3
%A _Christian G. Bower_
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