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 A280000 Number of free pure symmetric multifunctions in one symbol with n positions. 28
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS 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. LINKS Andrew Howroyd, Table of n, a(n) for n = 1..200 EXAMPLE 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 MATHEMATICA 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] PROG (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 CROSSREFS Cf. A005043 (non-symmetric case), A279944. Sequence in context: A135019 A141685 A017921 * A241097 A267725 A161762 Adjacent sequences: A279997 A279998 A279999 * A280001 A280002 A280003 KEYWORD nonn AUTHOR Gus Wiseman, Dec 24 2016 STATUS approved

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Last modified July 23 23:47 EDT 2024. Contains 374575 sequences. (Running on oeis4.)