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A124423 Number of partitions of the set {1,2,...,n} having no blocks that contain only even entries. 8
1, 1, 1, 3, 5, 22, 52, 283, 855, 5451, 19921, 144074, 614866, 4941987, 24040451, 211648665, 1152972925, 10998989896, 66200911138, 678600959525, 4465023867757, 48850849177703, 348383154017581, 4045835816532096, 31052765897026352, 381022649523561501 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Column 0 of A124422.

LINKS

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

FORMULA

a(n) = Q[n](1,0,1), 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_{j=0..ceiling(n/2)} Stirling2(ceiling(n/2),j) * j^floor(n/2). - Alois P. Heinz, Oct 23 2013

EXAMPLE

a(4) = 5 because we have 1234, 14|23, 1|234, 124|3 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: for n from 0 to 27 do Q[n]:=Q[n] od: seq(subs({t=1, s=0, x=1}, Q[n]), n=0..27);

# second Maple program:

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

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

MATHEMATICA

a[0] = a[1] = 1; a[n_] := Sum[StirlingS2[Ceiling[n/2], j]*j^Floor[n/2], {j, 0, Ceiling[n/2]}]; Table[a[n], {n, 0, 30}] (* Jean-Fran├žois Alcover, May 22 2015, after Alois P. Heinz *)

CROSSREFS

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

Sequence in context: A025098 A025117 A318076 * A209109 A212260 A178377

Adjacent sequences:  A124420 A124421 A124422 * A124424 A124425 A124426

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Oct 31 2006

STATUS

approved

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Last modified September 28 14:44 EDT 2022. Contains 357073 sequences. (Running on oeis4.)