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A102286 Total number of odd blocks in all partitions of n-set.

%I A102286 #24 Apr 10 2022 14:25:13
%S A102286 1,2,7,24,96,418,1989,10216,56275,330424,2057672,13532060,93633021,
%T A102286 679473694,5156626991,40824399712,336406367196,2879570703510,
%U A102286 25557841113625,234822774979908,2230107923204443,21861817965483016,220940261740238140,2299258336094622008
%N A102286 Total number of odd blocks in all partitions of n-set.
%C A102286 a(n) is also the number of set partitions of {1,2,...,n+1} in which the element 1 is in an even size block. - _Geoffrey Critzer_, Apr 02 2013
%H A102286 Alois P. Heinz, <a href="/A102286/b102286.txt">Table of n, a(n) for n = 1..575</a>
%F A102286 E.g.f: sinh(x)*exp(exp(x)-1).
%F A102286 a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n,2*k+1) * Bell(n-2*k-1). - _Ilya Gutkovskiy_, Apr 10 2022
%e A102286 a(3)=7 because we have (123), (1)/23, 12/(3), 13/(2), (1)/(2)/(3); the odd blocks are shown between parentheses.
%p A102286 G:=sinh(x)*exp(exp(x)-1): Gser:=series(G,x=0,30): seq(n!*coeff(Gser,x^n),n=1..25); # _Emeric Deutsch_
%p A102286 # second Maple program:
%p A102286 with(combinat):
%p A102286 b:= proc(n, i) option remember; `if`(n=0 or i=1, [1, n],
%p A102286        add((p->(p+[0, `if`(i::odd, j, 0)*p[1]]))(
%p A102286        b(n-i*j, i-1))*multinomial(n, n-i*j, i$j)/j!, j=0..n/i))
%p A102286     end:
%p A102286 a:= n-> b(n$2)[2]:
%p A102286 seq(a(n), n=1..30);  # _Alois P. Heinz_, Sep 16 2015
%t A102286 Range[0, nn]! CoefficientList[
%t A102286   D[Series[Exp[ (Cosh[x] - 1) + y Sinh[x]], {x, 0, nn}], y] /. y -> 1, x] (* _Geoffrey Critzer_, Aug 28 2012 *)
%t A102286 With[{nn=30},CoefficientList[Series[Sinh[x]Exp[Exp[x]-1],{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Jul 03 2021 *)
%Y A102286 Cf. A000110, A000296, A005493.
%K A102286 easy,nonn
%O A102286 1,2
%A A102286 _Vladeta Jovovic_, Feb 19 2005
%E A102286 More terms from _Emeric Deutsch_, Mar 04 2005