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Total number of even blocks in all partitions of n-set.
3

%I #14 Sep 16 2015 19:09:00

%S 0,1,3,13,55,256,1274,6791,38553,232171,1477355,9898780,69621864,

%T 512585529,3940556611,31560327945,262805569159,2271094695388,

%U 20333574916690,188322882941471,1801737999086129,17783472151154007,180866601699482803,1893373126840572056

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

%H Alois P. Heinz, <a href="/A102287/b102287.txt">Table of n, a(n) for n = 1..575</a>

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

%e a(3)=3 because in the 5 (=A000110(3)) partitions 123, (12)/3, (13)/2, 1/(23) and 1/2/3 of {1,2,3} we have 3 blocks of even size (shown between parentheses).

%p G:=(cosh(x)-1)*exp(exp(x)-1): Gser:=series(G,x=0,28): seq(n!*coeff(Gser,x^n),n=1..25); # _Emeric Deutsch_, Jun 22 2005

%p # second Maple program:

%p with(combinat):

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

%p add((p->(p+[0, `if`(i::odd, 0, j)*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[y (Cosh[x] - 1) + Sinh[x]], {x, 0, nn}], y] /. y -> 1, x] (* _Geoffrey Critzer_, Aug 28 2012 *)

%Y Cf. A005493, A000296.

%K easy,nonn

%O 1,3

%A _Vladeta Jovovic_, Feb 19 2005

%E More terms from _Emeric Deutsch_, Jun 22 2005