Revision History for A317652
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#9 by Alois P. Heinz at Tue Aug 28 20:20:43 EDT 2018
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#8 by Andrew Howroyd at Tue Aug 28 19:50:29 EDT 2018
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#7 by Andrew Howroyd at Tue Aug 28 19:23:41 EDT 2018
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| DATA
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1, 1, 2, 6, 22, 93, 421, 2010, 9926, 50357, 260728, 1372436, 7321982, 39504181, 215168221, 1181540841, 6534058589, 36357935615, 203414689462, 1143589234086, 6457159029573, 36602333187792, 208214459462774, 1188252476400972, 6801133579291811, 39032172166792887
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| LINKS
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Andrew Howroyd, <a href="/A317652/b317652.txt">Table of n, a(n) for n = 0..200</a>
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| PROG
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(PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
seq(n)={my(v=[]); for(n=1, n, my(t=EulerT(v)); v=concat(v, 1 + sum(k=1, n-1, v[k]*t[n-k]))); concat([1], v)} \\ Andrew Howroyd, Aug 28 2018
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| KEYWORD
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nonn,more
nonn
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| EXTENSIONS
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Terms a(12) and beyond from Andrew Howroyd, Aug 28 2018
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| STATUS
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approved
editing
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#6 by Susanna Cuyler at Fri Aug 03 08:16:44 EDT 2018
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#5 by Gus Wiseman at Fri Aug 03 05:29:48 EDT 2018
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#4 by Gus Wiseman at Fri Aug 03 05:29:21 EDT 2018
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| EXAMPLE
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The a(4) = 22 free pure symmetric multifunctions:
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#3 by Gus Wiseman at Fri Aug 03 04:56:52 EDT 2018
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#2 by Gus Wiseman at Fri Aug 03 03:56:58 EDT 2018
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| NAME
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allocatedNumber of free pure symmetric multifunctions whose leaves are an integer forpartition Gusof Wisemann.
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| DATA
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1, 1, 2, 6, 22, 93, 421, 2010, 9926, 50357, 260728, 1372436
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| OFFSET
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0,3
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| COMMENTS
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A free pure symmetric multifunction f in EPSM is either (case 1) a positive integer, or (case 2) an expression of the form h[g_1, ..., g_k] where k > 0, h is in EPSM, each of the g_i for i = 1, ..., k is in EPSM, and for i < j we have g_i <= g_j under a canonical total ordering of EPSM, such as the Mathematica ordering of expressions.
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| EXAMPLE
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The a(4) = 22 free pure symmetric multifunctions:
1[1[1[1]]] 1[1[2]] 1[3] 2[2] 4
1[1[1][1]] 1[2[1]] 3[1]
1[1][1[1]] 2[1[1]]
1[1[1]][1] 1[1][2]
1[1][1][1] 1[2][1]
1[1[1,1]] 2[1][1]
1[1,1[1]] 1[1,2]
1[1][1,1] 2[1,1]
1[1,1][1]
1[1,1,1]
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| MATHEMATICA
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sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
exprUsing[m_]:=exprUsing[m]=If[Length[m]==0, {{}}, If[Length[m]==1, {First[m]}, Join@@Cases[Union[Table[PR[m[[s]], m[[Complement[Range[Length[m]], s]]]], {s, Take[Subsets[Range[Length[m]]], {2, -2}]}]], PR[h_, g_]:>Join@@Table[Apply@@@Tuples[{exprUsing[h], Union[Sort/@Tuples[exprUsing/@p]]}], {p, mps[g]}]]]];
Table[Sum[Length[exprUsing[y]], {y, IntegerPartitions[n]}], {n, 0, 6}]
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| CROSSREFS
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Cf. A001003, A052893, A053492, A277996, A279944, A280000.
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| KEYWORD
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allocated
nonn,more
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| AUTHOR
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Gus Wiseman, Aug 03 2018
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| STATUS
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approved
editing
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#1 by Gus Wiseman at Fri Aug 03 03:56:58 EDT 2018
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| NAME
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allocated for Gus Wiseman
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| KEYWORD
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allocated
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| STATUS
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approved
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