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A089005 Number of partitions of n-set with at least one even block.

%I A089005 #11 Nov 18 2020 06:50:50
%S A089005 0,1,3,10,40,166,749,3683,19275,107806,640970,4024912,26653653,
%T A089005 185401581,1350624721,10282222002,81592209580,673535269054,
%U A089005 5773214891137,51291776763863,471617190143567,4481375500319334,43947651280912186,444258975094335440
%N A089005 Number of partitions of n-set with at least one even block.
%H A089005 Alois P. Heinz, <a href="/A089005/b089005.txt">Table of n, a(n) for n = 1..500</a>
%F A089005 E.g.f.: exp(sinh(x))*(exp(cosh(x)-1)-1).
%p A089005 with(combinat):
%p A089005 b:= proc(n, i, t) option remember; `if`(n=0, t, `if`(i<1, 0,
%p A089005        add(multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1,
%p A089005        max(t, `if`(j=0, 0, 1-irem(i, 2)))), j=0..n/i)))
%p A089005     end:
%p A089005 a:= n-> b(n$2, 0):
%p A089005 seq(a(n), n=1..30);
%t A089005 multinomial[n_, k_List] := n!/Times @@ (k!);
%t A089005 b[n_, i_, t_] := b[n, i, t] = If[n == 0, t, If[i<1, 0, Sum[multinomial[n, {n - i j} ~Join~ Table[i, {j}]]/j! b[n - i j, i - 1, Max[t, If[j == 0, 0, 1 - Mod[i, 2]]]], {j, 0, n/i}]]];
%t A089005 a[n_] := b[n, n, 0];
%t A089005 Array[a, 30] (* _Jean-François Alcover_, Nov 18 2020, after Maple *)
%o A089005 (PARI) my(x='x+O('x^30)); concat(0, Vec(serlaplace(exp(sinh(x))*(exp(cosh(x)-1)-1))))
%Y A089005 Cf. A000110, A005046, A003724, A059838, A047967.
%K A089005 nonn
%O A089005 1,3
%A A089005 _Vladeta Jovovic_, Nov 02 2003