A124425 Number of partitions of the set {1,2,...,n} having no blocks with all entries of the same parity.

1, 0, 1, 1, 3, 7, 25, 79, 339, 1351, 6721, 31831, 179643, 979567, 6166105, 37852039, 262308819, 1784037031, 13471274401, 100285059751, 818288740923, 6604485845167, 57836113793305, 502235849694679, 4693153430067699, 43572170967012871, 432360767273547841

Offset

0,5

Comments

Column 0 of A124424.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..400

Formula

a(n) = Q[n](0,0,1), 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_{k=0..floor(n/2)} Stirling2(floor(n/2),k)*Stirling2(ceiling(n/2),k)*k!. - Alois P. Heinz, Oct 24 2013

Example

a(4) = 3 because we have 1234, 14|23 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: seq(subs({t=0, s=0, x=1}, Q[n]), n=0..27);
# second Maple program:
a:= proc(n) local g, u; g:= floor(n/2); u:= ceil(n/2);
      add(Stirling2(g, k)*Stirling2(u, k)*k!, k=0..g)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Oct 24 2013

Mathematica

a[n_] := Module[{g=Floor[n/2], u=Ceiling[n/2]}, Sum[StirlingS2[g, k]*StirlingS2[u, k]*k!, {k, 0, g}]]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 26 2015, after Alois P. Heinz *)

Crossrefs

Cf. A000110, A124418, A124419, A124420, A124421, A124422, A124423, A124424.

Sequence in context: A130463 A148733 A148734 * A321606 A118398 A047974

Adjacent sequences: A124422 A124423 A124424 * A124426 A124427 A124428

Keyword

nonn

Author

Emeric Deutsch, Nov 01 2006

Status

approved