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A124421 Number of partitions of the set {1,2,...,n} having no blocks that contain only odd entries. 8

%I

%S 1,0,1,1,5,9,52,130,855,2707,19921,75771,614866,2717570,24040451,

%T 120652827,1152972925,6460552857,66200911138,408845736040,

%U 4465023867757,30083964854141,348383154017581,2539795748336375,31052765897026352,243282175672281360

%N Number of partitions of the set {1,2,...,n} having no blocks that contain only odd entries.

%C Column 0 of A124420.

%H Alois P. Heinz, <a href="/A124421/b124421.txt">Table of n, a(n) for n = 0..300</a>

%F 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.

%F a(n) = Sum_{j=0..floor(n/2)} Stirling2(floor(n/2),j) * j^ceiling(n/2). - _Alois P. Heinz_, Oct 23 2013

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

%p 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);

%p # second Maple program:

%p a:= n-> add(Stirling2(floor(n/2), j)*j^ceil(n/2), j=0..floor(n/2)):

%p seq(a(n), n=0..30); # _Alois P. Heinz_, Oct 23 2013

%t 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-Fran├žois Alcover_, May 20 2015, after _Alois P. Heinz_ *)

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

%K nonn

%O 0,5

%A _Emeric Deutsch_, Oct 31 2006

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