A124423 Number of partitions of the set {1,2,...,n} having no blocks that contain only even entries.

1, 1, 1, 3, 5, 22, 52, 283, 855, 5451, 19921, 144074, 614866, 4941987, 24040451, 211648665, 1152972925, 10998989896, 66200911138, 678600959525, 4465023867757, 48850849177703, 348383154017581, 4045835816532096, 31052765897026352, 381022649523561501

Offset

0,4

Comments

Column 0 of A124422.

Links

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

Formula

a(n) = Q[n](1,0,1), 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..ceiling(n/2)} Stirling2(ceiling(n/2),j) * j^floor(n/2). - Alois P. Heinz, Oct 23 2013

Example

a(4) = 5 because we have 1234, 14|23, 1|234, 124|3 and 12|34.

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=1, s=0, x=1}, Q[n]), n=0..27);
# second Maple program:
a:= n-> add(Stirling2(ceil(n/2), j)*j^floor(n/2), j=0..ceil(n/2)):
seq(a(n), n=0..30);  # Alois P. Heinz, Oct 23 2013

Mathematica

a[0] = a[1] = 1; a[n_] := Sum[StirlingS2[Ceiling[n/2], j]*j^Floor[n/2], {j, 0, Ceiling[n/2]}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 22 2015, after Alois P. Heinz *)

Crossrefs

Cf. A000110, A124418, A124419, A124420, A124421, A124422.

Sequence in context: A025098 A025117 A318076 * A209109 A212260 A178377

Adjacent sequences: A124420 A124421 A124422 * A124424 A124425 A124426

Keyword

nonn

Author

Emeric Deutsch, Oct 31 2006

Status

approved