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A124421 Number of partitions of the set {1,2,...n} having no blocks that contain only odd entries. 8
1, 0, 1, 1, 5, 9, 52, 130, 855, 2707, 19921, 75771, 614866, 2717570, 24040451, 120652827, 1152972925, 6460552857, 66200911138, 408845736040, 4465023867757, 30083964854141, 348383154017581, 2539795748336375, 31052765897026352, 243282175672281360 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Column 0 of A124420.

LINKS

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

FORMULA

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.

a(n) = Sum_{j=0..floor(n/2)} stirling2(floor(n/2),j) * j^ceil(n/2). - Alois P. Heinz, Oct 23 2013

EXAMPLE

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

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=0, s=1, x=1}, Q[n]), n=0..27);

# second Maple program:

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

seq(a(n), n=0..30);  # Alois P. Heinz, Oct 23 2013

MATHEMATICA

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 *)

CROSSREFS

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

Sequence in context: A289909 A000324 A123817 * A262918 A283918 A284382

Adjacent sequences:  A124418 A124419 A124420 * A124422 A124423 A124424

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Oct 31 2006

STATUS

approved

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Last modified February 21 12:58 EST 2018. Contains 299411 sequences. (Running on oeis4.)