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

0, 1, 3, 10, 40, 166, 749, 3683, 19275, 107806, 640970, 4024912, 26653653, 185401581, 1350624721, 10282222002, 81592209580, 673535269054, 5773214891137, 51291776763863, 471617190143567, 4481375500319334, 43947651280912186, 444258975094335440

Offset: 1

Author

Vladeta Jovovic, Nov 02 2003

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..500

Crossrefs

Cf. A000110, A005046, A003724, A059838, A047967.

Programs

Maple

with(combinat):
b:= proc(n, i, t) option remember; `if`(n=0, t, `if`(i<1, 0,
       add(multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1,
       max(t, `if`(j=0, 0, 1-irem(i, 2)))), j=0..n/i)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=1..30);

Mathematica

multinomial[n_, k_List] := n!/Times @@ (k!);
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}]]];
a[n_] := b[n, n, 0];
Array[a, 30] (* Jean-François Alcover, Nov 18 2020, after Maple *)

PARI

my(x='x+O('x^30)); concat(0, Vec(serlaplace(exp(sinh(x))*(exp(cosh(x)-1)-1))))

Formula

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