A032012 - OEIS

%I #17 Sep 13 2018 18:32:51

%S 1,1,0,1,8,1,12,1,128,3025,260,7921,2048,78937,4760,2375101,138411008,

%T 9837697,588189972,96605425,7353141248,1752111145,151280741480,

%U 9294316285,12191175684608,1413604888888801,75955683963432,9022098736088101,1170150933402368

%N Number of ways to partition n labeled elements into sets of different odd sizes and order the sets.

%H Alois P. Heinz, <a href="/A032012/b032012.txt">Table of n, a(n) for n = 0..742</a>

%H C. G. Bower, <a href="/transforms2.html">Transforms (2)</a>

%F "AGJ" (ordered, elements, labeled) transform of 1, 0, 1, 0, ...(odds).

%p b:= proc(n, i, p) option remember;

%p       `if`(n=0, p!, `if`(i<1, 0, b(n, i-2, p)+

%p       `if`(i>n, 0, b(n-i, i-2, p+1)*binomial(n, i))))

%p     end:

%p a:= n-> b(n, n-1+irem(n, 2), 0):

%p seq(a(n), n=0..30);

%t b[n_, i_, p_] := b[n, i, p] = If[n==0, p!, If[i<1, 0, b[n, i-2, p] + If[i>n, 0, b[n-i, i-2, p+1]*Binomial[n, i]]]]; a[n_] := b[n, n-1+Mod[n, 2], 0]; Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Feb 27 2017, translated from Maple *)

%o (PARI) seq(n)=[subst(serlaplace(y^0*p),y,1) | p <- Vec(serlaplace(prod(k=1, ceil(n/2), 1 + x^(2*k-1)*y/(2*k-1)! + O(x*x^n))))] \\ _Andrew Howroyd_, Sep 13 2018

%K nonn

%O 0,5

%A _Christian G. Bower_

%E a(0)=1 prepended by _Alois P. Heinz_, May 11 2016