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A280000
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Number of free pure symmetric multifunctions in one symbol with n positions.
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28
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1, 0, 1, 1, 3, 5, 12, 25, 57, 128, 296, 688, 1618, 3839, 9170, 22065, 53370, 129807, 317080, 777887, 1915247, 4731932, 11726476, 29143123, 72614115, 181363151, 453975928, 1138697689, 2861607677, 7204169689
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OFFSET
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1,5
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COMMENTS
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A free pure symmetric multifunction (PSM) in one symbol x is either (case 1) the symbol x, or (case 2) an expression of the form h[g_1,...,g_k] where h is a PSM in x, each of the g_i for i=1..(k>0) is a PSM in x, and for i < j we have g_i <= g_j under a canonical total ordering such as the Mathematica ordering. The number of positions in a PSM is the number of brackets [...] plus the number of x's.
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LINKS
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EXAMPLE
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Sequence of free pure symmetric multifunctions (second column) together with their numbers of positions (first column) and j-numbers (third column, see A279944 for details) begins:
1 x 1
3 x[x] 2
4 x[x,x] 8
5 x[x][x] 3
5 x[x[x]] 4
5 x[x,x,x] 128
6 x[x,x][x] 12
6 x[x][x,x] 27
6 x[x,x[x]] 32
6 x[x,x,x,x] 32768
6 x[x[x,x]] 262144
7 x[x][x][x] 5
7 x[x[x]][x] 6
7 x[x][x[x]] 9
7 x[x[x][x]] 16
7 x[x[x[x]]] 64
7 x[x,x,x][x] 145
7 x[x,x][x,x] 1728
7 x[x,x,x[x]] 2048
7 x[x][x,x,x] 2187
7 x[x,x,x,x,x] 2147483648
7 x[x,x[x,x]] 137438953472
7 x[x[x,x,x]] 1378913...3030144
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MATHEMATICA
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multing[t_, n_]:=Array[(t+#-1)/#&, n, 1, Times];
a[n_]:=If[n===1, 1, Sum[a[k]*Sum[Product[multing[a[First[s]], Length[s]], {s, Split[p]}], {p, IntegerPartitions[n-k-1]}], {k, 1, n-2}]];
Array[a, 15]
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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=[1]); for(n=2, n, my(t=EulerT(v)); v=concat(v, sum(k=1, n-2, v[k]*t[n-k-1]))); v} \\ Andrew Howroyd, Aug 19 2018
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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