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 A124419 Number of partitions of the set {1,2,...n} having no blocks that contain both odd and even entries. 14
 1, 1, 1, 2, 4, 10, 25, 75, 225, 780, 2704, 10556, 41209, 178031, 769129, 3630780, 17139600, 87548580, 447195609, 2452523325, 13450200625, 78697155750, 460457244900, 2859220516290, 17754399678409, 116482516809889, 764214897046969, 5277304280371714 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Column 0 of A124418. LINKS Alois P. Heinz, Table of n, a(n) for n = 0..500 A. Dzhumadil’daev and D. Yeliussizov, Path decompositions of digraphs and their applications to Weyl algebra, arXiv preprint arXiv:1408.6764v1, 2014. [Version 1 contained many references to the OEIS, which were removed in Version 2. - N. J. A. Sloane, Mar 28 2015] Askar Dzhumadil’daev and Damir Yeliussizov, Walks, partitions, and normal ordering, Electronic Journal of Combinatorics, 22(4) (2015), #P4.10. FORMULA a(n) = Q[n](1,1,0), 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) = A000110(floor(n/2)) * A000110(ceiling(n/2)). - Alois P. Heinz, Oct 23 2013 EXAMPLE a(4) = 4 because we have 13|24, 1|24|3, 13|2|4 and 1|2|3|4. MAPLE Q[0]:=1: for n from 1 to 30 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 30 do Q[n]:=Q[n] od: seq(subs({t=1, s=1, x=0}, Q[n]), n=0..30); # second Maple program: with(combinat): a:= n-> bell(floor(n/2))*bell(ceil(n/2)): seq(a(n), n=0..30);  # Alois P. Heinz, Oct 23 2013 MATHEMATICA a[n_] := BellB[Floor[n/2]]*BellB[Ceiling[n/2]]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 20 2015, after Alois P. Heinz *) CROSSREFS Cf. A000110, A124418, A124420, A124421, A124422, A124423, A274310. Column k=2 of A275069. Sequence in context: A206289 A148094 A148095 * A148096 A006901 A123422 Adjacent sequences:  A124416 A124417 A124418 * A124420 A124421 A124422 KEYWORD nonn AUTHOR Emeric Deutsch, Oct 31 2006 STATUS approved

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