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

%I A124425 #19 Jun 24 2019 22:23:46
%S A124425 1,0,1,1,3,7,25,79,339,1351,6721,31831,179643,979567,6166105,37852039,
%T A124425 262308819,1784037031,13471274401,100285059751,818288740923,
%U A124425 6604485845167,57836113793305,502235849694679,4693153430067699,43572170967012871,432360767273547841
%N A124425 Number of partitions of the set {1,2,...,n} having no blocks with all entries of the same parity.
%C A124425 Column 0 of A124424.
%H A124425 Alois P. Heinz, <a href="/A124425/b124425.txt">Table of n, a(n) for n = 0..400</a>
%F A124425 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.
%F A124425 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
%e A124425 a(4) = 3 because we have 1234, 14|23 and 12|34.
%p A124425 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);
%p A124425 # second Maple program:
%p A124425 a:= proc(n) local g, u; g:= floor(n/2); u:= ceil(n/2);
%p A124425       add(Stirling2(g, k)*Stirling2(u, k)*k!, k=0..g)
%p A124425     end:
%p A124425 seq(a(n), n=0..30);  # _Alois P. Heinz_, Oct 24 2013
%t A124425 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_ *)
%Y A124425 Cf. A000110, A124418, A124419, A124420, A124421, A124422, A124423, A124424.
%K A124425 nonn
%O A124425 0,5
%A A124425 _Emeric Deutsch_, Nov 01 2006