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A124425 Number of partitions of the set {1,2,...n} having no blocks with all entries of the same parity. 2
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 (list; graph; refs; listen; history; text; internal format)
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(ceil(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-Fran├žois 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

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Last modified January 18 16:40 EST 2019. Contains 319271 sequences. (Running on oeis4.)