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A032109 "BIJ" (reversible, indistinct, labeled) transform of 1,1,1,1,... 3
1, 1, 2, 7, 38, 271, 2342, 23647, 272918, 3543631, 51123782, 811316287, 14045783798, 263429174191, 5320671485222, 115141595488927, 2657827340990678, 65185383514567951, 1692767331628422662, 46400793659664205567, 1338843898122192101558, 40562412499252036940911 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
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)
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
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 *)
PROG
(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
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.
Sequence in context: A209006 A168492 A346646 * A368232 A337026 A088792
KEYWORD
nonn
AUTHOR
STATUS
approved

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Last modified March 4 05:23 EST 2024. Contains 370522 sequences. (Running on oeis4.)