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

0,5

Comments

Column 0 of A124420.

Links

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

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

Example

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

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 *)

Crossrefs

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

Bisection gives A108459 (even part).

Sequence in context: A000324 A123817 A369155 * A262918 A283918 A284382

Adjacent sequences: A124418 A124419 A124420 * A124422 A124423 A124424

Keyword

nonn

Author

Emeric Deutsch, Oct 31 2006

Status

approved