A124421 Number of partitions of the set {1,2,...,n} having no blocks that contain only odd entries.

1, 0, 1, 1, 5, 9, 52, 130, 855, 2707, 19921, 75771, 614866, 2717570, 24040451, 120652827, 1152972925, 6460552857, 66200911138, 408845736040, 4465023867757, 30083964854141, 348383154017581, 2539795748336375, 31052765897026352, 243282175672281360

Offset: 0

Author

Emeric Deutsch, Oct 31 2006

Keywords

• nonn

Comments

Column 0 of A124420.

Examples

			a(4) = 5 because we have 1234, 134|2, 14|23, 12|34 and 123|4.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A000110, A124418, A124419, A124420, A124422, A124423.

Bisection gives A108459 (even part).

Programs

Maple

Q[0]:=1: for n from 1 to 27 do if n mod 2 = 1 then Q[n]:=expand(t*diff(Q[n-1],t)+x*diff(Q[n-1],s)+x*diff(Q[n-1],x)+t*Q[n-1]) else Q[n]:=expand(x*diff(Q[n-1],t)+s*diff(Q[n-1],s)+x*diff(Q[n-1],x)+s*Q[n-1]) fi od: for n from 0 to 27 do Q[n]:=Q[n] od: seq(subs({t=0,s=1,x=1},Q[n]),n=0..27);
# second Maple program:
a:= n-> add(Stirling2(floor(n/2), j)*j^ceil(n/2), j=0..floor(n/2)):
seq(a(n), n=0..30);  # Alois P. Heinz, Oct 23 2013

Mathematica

a[0] = 1; a[n_] := Sum[StirlingS2[Floor[n/2], j]*j^Ceiling[n/2], {j, 0, Floor[n/2]}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 20 2015, after Alois P. Heinz *)

Formula

a(n) = Q[n](0,1,1), where the polynomials Q[n]=Q[n](t,s,x) are defined by Q[0]=1; Q[n]=t*dQ[n-1]/dt + x*dQ[n-1]/ds + x*dQ[n-1]/dx + t*Q[n-1] if n is odd and Q[n]=x*dQ[n-1]/dt + s*dQ[n-1]/ds + x*dQ[n-1]/dx + s*Q[n-1] if n is even.

a(n) = Sum_{j=0..floor(n/2)} Stirling2(floor(n/2),j) * j^ceiling(n/2). - Alois P. Heinz, Oct 23 2013