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Total number of odd blocks in all partitions of n-set.
5

%I #24 Apr 10 2022 14:25:13

%S 1,2,7,24,96,418,1989,10216,56275,330424,2057672,13532060,93633021,

%T 679473694,5156626991,40824399712,336406367196,2879570703510,

%U 25557841113625,234822774979908,2230107923204443,21861817965483016,220940261740238140,2299258336094622008

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

%C 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 Alois P. Heinz, <a href="/A102286/b102286.txt">Table of n, a(n) for n = 1..575</a>

%F E.g.f: sinh(x)*exp(exp(x)-1).

%F 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 a(3)=7 because we have (123), (1)/23, 12/(3), 13/(2), (1)/(2)/(3); the odd blocks are shown between parentheses.

%p 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 # second Maple program:

%p with(combinat):

%p b:= proc(n, i) option remember; `if`(n=0 or i=1, [1, n],

%p add((p->(p+[0, `if`(i::odd, j, 0)*p[1]]))(

%p b(n-i*j, i-1))*multinomial(n, n-i*j, i$j)/j!, j=0..n/i))

%p end:

%p a:= n-> b(n$2)[2]:

%p seq(a(n), n=1..30); # _Alois P. Heinz_, Sep 16 2015

%t Range[0, nn]! CoefficientList[

%t D[Series[Exp[ (Cosh[x] - 1) + y Sinh[x]], {x, 0, nn}], y] /. y -> 1, x] (* _Geoffrey Critzer_, Aug 28 2012 *)

%t 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 Cf. A000110, A000296, A005493.

%K easy,nonn

%O 1,2

%A _Vladeta Jovovic_, Feb 19 2005

%E More terms from _Emeric Deutsch_, Mar 04 2005