A115277 Number of partitions of {1,...,n} into blocks such that no even sized block is repeated.

1, 1, 2, 5, 12, 37, 143, 562, 2320, 10941, 54865, 283890, 1604155, 9558226, 58668223, 384572975, 2631778832, 18576630237, 137919691717, 1060303298138, 8415786131309, 69538205444478, 591734670548037, 5194542789203877, 47127033586211659, 438972204436025198

Offset: 0

Author

Christian G. Bower, Jan 18 2006

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A001935, A115275, A115276, A115278.

Programs

Maple

with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
       multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1), j=0..min(
       `if`(irem(i, 2)=0, 1, n), n/i))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 08 2015

Mathematica

multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[multinomial[n, Join[{n-i*j}, Array[i&, j]]]/j! * b[n-i*j, i-1], {j, 0, Min[If[Mod[i, 2]==0, 1, n], n/i]}]]]; a[n_] :=  b[n, n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Oct 25 2015, after Alois P. Heinz *)

Formula

E.g.f.: exp(sinh(x)) * Product {m >= 1} (1+x^(2*m)/(2*m)!).