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A032310
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Number of ways to partition n labeled elements into sets of odd sizes, with all sizes different.
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6
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1, 1, 0, 1, 4, 1, 6, 1, 64, 505, 130, 1321, 1024, 13157, 2380, 395851, 5782144, 1639617, 24545706, 16100905, 306621184, 292018525, 6304002100, 1549052715, 507969498304, 11794047630801, 3164830777316, 75389026652551, 48756350408224, 1240389053007865
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OFFSET
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0,5
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LINKS
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FORMULA
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"EGJ" (unordered, element, labeled) transform of 1, 0, 1, 0... (odds)
E.g.f.: Product_{k>0} (1+x^(2*k-1)/(2*k-1)!). - Vladeta Jovovic, Jan 16 2004
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MAPLE
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with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-2), j=0..min(1, n/i))))
end:
a:= n-> b(n, iquo(n+1, 2)*2-1):
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MATHEMATICA
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multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[multinomial[n, Join[{n-i*j}, Array[i&, j]]]/j!*b[n - i*j, i-2], {j, 0, Min[1, n/i]}]]]; a[n_] := b[n, Quotient[n+1, 2]*2-1]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 05 2017, after Alois P. Heinz *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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