A052893 -id:A052893 - cp's OEIS frontend

A277996 Number of free pure symmetric multifunctions (with empty expressions allowed) with one atom and n positions.

1, 1, 2, 5, 13, 36, 102, 299, 892, 2713, 8364, 26108, 82310, 261804, 838961, 2706336, 8780725, 28636157, 93818641, 308641277, 1019140129, 3376604826, 11221805968, 37399728251, 124967677989, 418564867751, 1405030366113, 4726036692421, 15927027834163, 53770343259613

Offset: 1

Gus Wiseman, Dec 24 2016

nonn

Also the number of distinct orderless Mathematica expressions with one atom and n positions.

			The a(5)=13 Mathematica expressions are:
x[x,x,x]
x[x,x][]   x[x][x]   x[][x,x]  x[x,x[]]  x[x[x]]
x[x][][]   x[][x][]  x[][][x]  x[x[]][]  x[][x[]]  x[x[][]]
x[][][][]

Andrew Howroyd, Table of n, a(n) for n = 1..200
Mathematica Reference, Orderless.

Cf. A000108, A001003, A005043, A052893, A279944, A280000, A317658.

multing[t_,n_]:=Array[(t+#-1)/#&,n,1,Times];
a[n_]:=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-1}]];
Array[a,30]

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, v[n-1] + sum(k=1, n-2, v[k]*t[n-k-1]))); v} \\ Andrew Howroyd, Aug 19 2018

From Ilya Gutkovskiy, Apr 30 2019: (Start)
G.f. A(x) satisfies: A(x) = x * (1 + A(x) * exp(Sum_{k>=1} A(x^k)/k)).
G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x * (1 + (Sum_{n>=1} a(n)*x^n) * Product_{n>=1} 1/(1 - x^n)^a(n)). (End)

A317658 Number of positions in the n-th free pure symmetric multifunction (with empty expressions allowed) with one atom.

Original entry on oeis.org

1, 2, 3, 3, 4, 4, 5, 4, 4, 5, 6, 5, 5, 6, 7, 4, 6, 6, 7, 8, 5, 7, 7, 8, 5, 9, 5, 6, 8, 8, 9, 5, 6, 10, 6, 5, 7, 9, 9, 10, 6, 7, 11, 7, 6, 8, 10, 10, 6, 11, 7, 8, 12, 8, 7, 9, 11, 11, 7, 12, 8, 9, 13, 5, 9, 8, 10, 12, 12, 8, 13, 9, 10, 14, 6, 10, 9, 11, 13, 13

Offset: 1

Gus Wiseman, Aug 03 2018

nonn

Given a positive integer n > 1 we construct a unique free pure symmetric multifunction e(n) by expressing n as a power of a number that is not a perfect power to a product of prime numbers: n = rad(x)^(prime(y_1) * ... * prime(y_k)) where rad = A007916. Then e(n) = e(x)[e(y_1), ..., e(y_k)].

Also the number of positions in the orderless Mathematica expression with e-number n.

			The first twenty Mathematica expressions:
   1: o
   2: o[]
   3: o[][]
   4: o[o]
   5: o[][][]
   6: o[o][]
   7: o[][][][]
   8: o[o[]]
   9: o[][o]
  10: o[o][][]
  11: o[][][][][]
  12: o[o[]][]
  13: o[][o][]
  14: o[o][][][]
  15: o[][][][][][]
  16: o[o,o]
  17: o[o[]][][]
  18: o[][o][][]
  19: o[o][][][][]
  20: o[][][][][][][]

Mathematica Reference, Orderless

First differs from A277615 at a(128) = 5, A277615(128) = 6.
Cf. A001003, A052893, A053492, A255906, A277996, A279944, A280000.
Cf. A317652, A317653, A317654, A317655, A317656.

nn=100;
radQ[n_]:=If[n===1,False,GCD@@FactorInteger[n][[All,2]]===1];
rad[n_]:=rad[n]=If[n===0,1,NestWhile[#+1&,rad[n-1]+1,Not[radQ[#]]&]];
Clear[radPi];Set@@@Array[radPi[rad[#]]==#&,nn];
exp[n_]:=If[n===1,x,With[{g=GCD@@FactorInteger[n][[All,2]]},Apply[exp[radPi[Power[n,1/g]]],exp/@Flatten[Cases[FactorInteger[g],{p_?PrimeQ,k_}:>ConstantArray[PrimePi[p],k]]]]]];
Table[exp[n],{n,1,nn}]

a(rad(x)^(prime(y_1) * ... * prime(y_k))) = a(x) + a(y_1) + ... + a(y_k).
e(2^(2^n)) = o[o,...,o].
e(2^prime(2^prime(2^...))) = o[o[...o[o]]].
e(rad(rad(rad(...)^2)^2)^2) = o[o][o]...[o].

A317875 Number of achiral free pure multifunctions with n unlabeled leaves.

Original entry on oeis.org

1, 1, 3, 9, 30, 102, 369, 1362, 5181, 20064, 79035, 315366, 1272789, 5185080, 21296196, 88083993, 366584253, 1533953100, 6449904138, 27238006971, 115475933202, 491293053093, 2096930378415, 8976370298886, 38528771056425, 165784567505325

Offset: 1

Gus Wiseman, Aug 09 2018

nonn

An achiral free pure multifunction is either (case 1) the leaf symbol "o", or (case 2) a nonempty expression of the form h[g, ..., g], where h and g are both achiral free pure multifunctions.

			The first 4 terms count the following multifunctions.
o,
o[o],
o[o,o], o[o[o]], o[o][o],
o[o,o,o], o[o[o][o]], o[o[o[o]]], o[o[o,o]], o[o][o,o], o[o][o[o]], o[o][o][o], o[o,o][o], o[o[o]][o].

Andrew Howroyd, Table of n, a(n) for n = 1..200

Cf. A001003, A001678, A002033, A003238, A052893, A053492, A067824, A167865, A214577, A277996, A280000, A317853.
Cf. A317876, A317877, A317878, A317879, A317880, A317881.
Cf. A317882, A317883, A317884, A317885.

a[n_]:=If[n==1,1,Sum[a[n-k]*Sum[a[d],{d,Divisors[k]}],{k,n-1}]];
Array[a,12]

seq(n)={my(p=O(x)); for(n=1, n, p = x + p*(sum(k=1, n-1, subst(p + O(x^(n\k+1)), x, x^k)) ) + O(x*x^n)); Vec(p)} \\ Andrew Howroyd, Aug 19 2018

seq(n)={my(v=vector(n)); v[1]=1; for(n=2, #v, v[n]=sum(i=1, n-1, v[i]*sumdiv(n-i, d, v[d]))); v} \\ Andrew Howroyd, Aug 19 2018

a(1) = 1; a(n > 1) = Sum_{0 < k < n} a(n - k) * Sum_{d|k} a(d).
From Ilya Gutkovskiy, Apr 30 2019: (Start)
G.f. A(x) satisfies: A(x) = x + A(x) * Sum_{k>=1} A(x^k).
G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x + (Sum_{n>=1} a(n)*x^n) * (Sum_{n>=1} a(n)*x^n/(1 - x^n)). (End)

A317654 Number of free pure symmetric multifunctions whose leaves are a strongly normal multiset of size n.

Original entry on oeis.org

1, 3, 26, 375, 6696, 159837, 4389226, 144915350, 5377002075, 227624621051, 10632808475596, 550932945236121, 31062550998284221, 1907051034025848314, 126052420069459211076, 8956882232940915920404, 679298518935625486287703, 54868537321267493152151502, 4696952405203792017289469056

Offset: 1

Gus Wiseman, Aug 03 2018

nonn

A multiset is strongly normal if it spans an initial interval of positive integers with weakly decreasing multiplicities. 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.

			The a(3) = 26 free pure symmetric multifunctions:
1[1[1]], 1[1,1], 1[1][1],
1[1[2]], 1[2[1]], 1[1,2], 2[1[1]], 2[1,1], 1[1][2], 1[2][1], 2[1][1],
1[2[3]], 1[3[2]], 1[2,3], 2[1[3]], 2[3[1]], 2[1,3], 3[1[2]], 3[2[1]], 3[1,2], 1[2][3], 2[1][3], 1[3][2], 3[1][2], 2[3][1], 3[2][1].

Cf. A001003, A052893, A053492, A255906, A277996, A279944, A280000.

Cf. A317652, A317653, A317655, A317656, A317658.

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]}]]]];
got[y_]:=Join@@Table[Table[i,{y[[i]]}],{i,Range[Length[y]]}];
Table[Sum[Length[exprUsing[got[y]]],{y,IntegerPartitions[n]}],{n,6}]

\\ See links in A339645 for combinatorial species functions.
cycleIndexSeries(n)={my(p=O(x)); for(n=1, n, p = x*sv(1) + p*(sExp(p)-1)); p}
StronglyNormalLabelingsSeq(cycleIndexSeries(15)) \\ Andrew Howroyd, Jan 01 2021

Terms a(8) and beyond from Andrew Howroyd, Jan 01 2021

A317652 Number of free pure symmetric multifunctions whose leaves are an integer partition of n.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Aug 03 2018

nonn

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.

			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]

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A001003, A052893, A053492, A277996, A279944, A280000.

Cf. A317653, A317654, A317655, A317656, A317658.

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}]

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

Terms a(12) and beyond from Andrew Howroyd, Aug 28 2018

A317653 Number of free pure symmetric multifunctions whose leaves are a normal multiset of size n.

Original entry on oeis.org

1, 3, 34, 602, 14872, 472138, 18323359, 840503724, 44489123726, 2668985463839, 178960530393633, 13263068003965046, 1076580864432281157, 94987639225399100006, 9051397653144246683937, 926407121115738135640677, 101357200280211387377806719, 11804887470887800839909147484

Offset: 1

Gus Wiseman, Aug 03 2018

nonn

A multiset is normal if it spans an initial interval of positive integers. 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.

			The a(3) = 34 free pure symmetric multifunctions:
1[1[1]], 1[1,1], 1[1][1],
1[2[2]], 1[2,2], 2[1[2]], 2[2[1]], 2[1,2], 1[2][2], 2[1][2], 2[2][1],
1[1[2]], 1[2[1]], 1[1,2], 2[1[1]], 2[1,1], 1[1][2], 1[2][1], 2[1][1],
1[2[3]], 1[3[2]], 1[2,3], 2[1[3]], 2[3[1]], 2[1,3], 3[1[2]], 3[2[1]], 3[1,2], 1[2][3], 2[1][3], 1[3][2], 3[1][2], 2[3][1], 3[2][1].

Andrew Howroyd, Table of n, a(n) for n = 1..200

Cf. A001003, A052893, A053492, A255906, A277996, A279944, A280000.

Cf. A317652, A317654, A317655, A317656, A317658.

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]}]]]];
got[y_]:=Join@@Table[Table[i,{y[[i]]}],{i,Range[Length[y]]}];
Table[Sum[Length[exprUsing[got[y]]],{y,Join@@Permutations/@IntegerPartitions[n]}],{n,6}]

\\ here R(n,1) is A052893.
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
R(n,k)={my(v=[k]); for(n=2, n, my(t=EulerT(v)); v=concat(v, sum(k=1, n-1, v[k]*t[n-k]))); v}
seq(n)={sum(k=1, n, R(n,k)*sum(r=k, n, binomial(r,k)*(-1)^(r-k)) )} \\ Andrew Howroyd, Sep 14 2018

Terms a(8) and beyond from Andrew Howroyd, Sep 14 2018

A317876 Number of free pure symmetric identity multifunctions (with empty expressions allowed) with one atom and n positions.

Original entry on oeis.org

1, 1, 2, 4, 10, 25, 67, 184, 519, 1489, 4342, 12812, 38207, 114934, 348397, 1063050, 3262588, 10064645, 31190985, 97061431, 303165207, 950115502, 2986817742, 9415920424, 29760442192, 94286758293, 299377379027, 952521579944, 3036380284111, 9696325863803

Offset: 1

Gus Wiseman, Aug 09 2018

nonn

A free pure symmetric identity multifunction (with empty expressions allowed) (FOI) is either (case 1) the leaf symbol "o", or (case 2) a possibly empty expression of the form h[g_1, ..., g_k] where h is an FOI, each of the g_i for i = 1, ..., k >= 0 is an FOI, and for i < j we have g_i < g_j under a canonical total ordering such as the Mathematica ordering of expressions. The number of positions in an FOI is the number of brackets [...] plus the number of o's.

Also the number of free orderless identity Mathematica expressions with one atom and n positions.

			The a(5) = 10 FOIs:
  o[o[o]]
  o[o][o]
  o[o[][]]
  o[o,o[]]
  o[][o[]]
  o[][][o]
  o[o[]][]
  o[][o][]
  o[o][][]
  o[][][][]

Andrew Howroyd, Table of n, a(n) for n = 1..200

Cf. A000081, A004111, A052893, A053492, A277996, A280000, A317652, A317653, A317654, A317875.

Cf. A317877, A317878, A317879, A317880, A317881.

allIdExpr[n_]:=If[n==1,{"o"},Join@@Cases[Table[PR[k,n-k-1],{k,n-1}],PR[h_,g_]:>Join@@Table[Apply@@@Tuples[{allIdExpr[h],Select[Union[Sort/@Tuples[allIdExpr/@p]],UnsameQ@@#&]}],{p,IntegerPartitions[g]}]]];
Table[Length[allIdExpr[n]],{n,12}]

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
seq(n)={my(v=[1]); for(n=2, n, my(t=WeighT(v)); v=concat(v, v[n-1] + sum(k=1, n-2, v[k]*t[n-k-1]))); v} \\ Andrew Howroyd, Aug 19 2018

From Ilya Gutkovskiy, Apr 30 2019: (Start)
G.f. A(x) satisfies: A(x) = x * (1 + A(x) * exp(Sum_{k>=1} (-1)^(k+1)*A(x^k)/k)).
G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x * (1 + (Sum_{n>=1} a(n)*x^n) * Product_{n>=1} (1 + x^n)^a(n)). (End)

Terms a(16) and beyond from Andrew Howroyd, Aug 19 2018

A317877 Number of free pure identity multifunctions with one atom and n positions.

Original entry on oeis.org

1, 0, 1, 0, 2, 2, 5, 10, 18, 46, 94, 212, 476, 1058, 2441, 5564, 12880, 29920, 69620, 163220, 383376, 904114, 2139592, 5074784, 12074152, 28789112, 68803148, 164779064, 395373108, 950416330, 2288438591, 5518864858, 13329183894, 32237132814, 78069124640

Offset: 1

Gus Wiseman, Aug 09 2018

nonn

A free pure identity multifunction (PIM) is either (case 1) the leaf symbol "o", or (case 2) an expression of the form h[g_1, ..., g_k] where h is a PIM, each of the g_i for i = 1, ..., k > 0 is a PIM, and for i != j we have g_i != g_j. The number of positions in a PIM is the number of brackets [...] plus the number of o's.

			The a(8) = 10 PIMs:
  o[o[o[o],o]]
  o[o[o,o[o]]]
  o[o[o[o]],o]
  o[o[o][o],o]
  o[o,o[o[o]]]
  o[o,o[o][o]]
  o[o][o[o],o]
  o[o][o,o[o]]
  o[o[o],o][o]
  o[o,o[o]][o]

Andrew Howroyd, Table of n, a(n) for n = 1..200

Cf. A000081, A001003, A003238, A004111, A052893, A053492, A277996, A280000, A317875.

Cf. A317876, A317878, A317879, A317880, A317881.

allIdPMF[n_]:=If[n==1,{"o"},Join@@Cases[Table[PR[k,n-k-1],{k,n-2}],PR[h_,g_]:>Join@@Table[Apply@@@Tuples[{allIdPMF[h],Select[Tuples[allIdPMF/@p],UnsameQ@@#&]}],{p,Join@@Permutations/@IntegerPartitions[g]}]]];
Table[Length[allIdPMF[n]],{n,12}]

seq(n)={my(v=vector(n)); v[1]=1; for(n=2, n, my(p=prod(k=1, n, 1 + sum(i=1, n\k, binomial(v[k], i)*x^(i*k)*y^i) + O(x*x^n))); v[n]=sum(k=1, n-2, v[n-k-1]*subst(serlaplace(y^0*polcoef(p, k)), y, 1))); v} \\ Andrew Howroyd, Sep 01 2018

Terms a(13) and beyond from Andrew Howroyd, Sep 01 2018

A317878 Number of free pure symmetric identity multifunctions with one atom and n positions.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 5, 5, 15, 23, 54, 98, 212, 420, 886, 1822, 3838, 8046, 17029, 36097, 76889, 164245, 351971, 756341, 1629389, 3518643, 7614717, 16512962, 35875986, 78082171, 170219300, 371651968, 812624721, 1779240627, 3900634491, 8561723769, 18814112811

Offset: 1

Gus Wiseman, Aug 09 2018

nonn

A free pure symmetric identity multifunction (SIM) is either (case 1) the leaf symbol "o", or (case 2) an expression of the form h[g_1, ..., g_k] where h is a SIM, each of the g_i for i = 1, ..., k > 0 is a SIM, and for i < j we have g_i < g_j under a canonical total ordering such as the Mathematica ordering of expressions. The number of positions in a SIM is the number of brackets [...] plus the number of o's.

			The a(8) = 5 SIMs:
  o[o[o,o[o]]]
  o[o,o[o[o]]]
  o[o,o[o][o]]
  o[o][o,o[o]]
  o[o,o[o]][o]

Andrew Howroyd, Table of n, a(n) for n = 1..200

Cf. A000081, A003238, A004111, A052893, A053492, A277996, A280000, A317875.

Cf. A317876, A317877, A317879, A317880, A317881.

allIdPMFOL[n_]:=If[n==1,{"o"},Join@@Cases[Table[PR[k,n-k-1],{k,n-2}],PR[h_,g_]:>Join@@Table[Apply@@@Tuples[{allIdPMFOL[h],Select[Union[Sort/@Tuples[allIdPMFOL/@p]],UnsameQ@@#&]}],{p,IntegerPartitions[g]}]]];
Table[Length[allIdPMFOL[n]],{n,12}]

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
seq(n)={my(v=[1]); for(n=2, n, my(t=WeighT(v)); v=concat(v, sum(k=1, n-2, v[k]*t[n-k-1]))); v} \\ Andrew Howroyd, Aug 19 2018

Terms a(13) and beyond from Andrew Howroyd, Aug 19 2018

A317880 Number of series-reduced free pure symmetric identity multifunctions (with empty expressions allowed) with one atom and n positions.

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 8, 16, 33, 70, 152, 333, 735, 1635, 3668, 8285, 18823, 42970, 98535, 226870, 524290, 1215641, 2827203, 6593432, 15416197, 36129894, 84860282, 199719932, 470930802, 1112388190, 2631903295, 6236669381, 14800078408, 35169529363, 83680908692

Offset: 1

Gus Wiseman, Aug 09 2018

nonn

A series-reduced free pure symmetric identity multifunction (with empty expressions allowed) (SROI) is either (case 1) the leaf symbol "o", or (case 2) a possibly empty expression of the form h[g_1, ..., g_k] where h is an SROI, k is an integer greater than or equal to 0 but not equal to 1, each of the g_i for i = 1, ..., k is an SROI, and for i < j we have g_i < g_j under a canonical total ordering such as the Mathematica ordering of expressions. The number of positions in an SROI is the number of brackets [...] plus the number of o's.

Also the number of series-reduced orderless identity Mathematica expressions with one atom and n positions.

			The a(7) = 8 SROIs:
  o[o,o[][][]]
  o[o[],o[][]]
  o[][o,o[][]]
  o[][][o,o[]]
  o[o,o[][]][]
  o[][o,o[]][]
  o[o,o[]][][]
  o[][][][][][]

Andrew Howroyd, Table of n, a(n) for n = 1..200

Cf. A001003, A001678, A052893, A053492, A067824, A167865, A277996, A280000, A317875.

Cf. A317876, A317877, A317878, A317879, A317880, A317881.

allIdExprSR[n_]:=If[n==1,{"o"},Join@@Cases[Table[PR[k,n-k-1],{k,n-1}],PR[h_,g_]:>Join@@Table[Apply@@@Tuples[{allIdExprSR[h],Select[Union[Sort/@Tuples[allIdExprSR/@p]],UnsameQ@@#&]}],{p,If[g==0,{{}},Rest[IntegerPartitions[g]]]}]]];
Table[Length[allIdExprSR[n]],{n,12}]

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
seq(n)={my(v=[1]); for(n=2, n, my(t=WeighT(v)-v); v=concat(v, v[n-1] + sum(k=1, n-2, v[k]*t[n-k-1]))); v} \\ Andrew Howroyd, Aug 19 2018

Terms a(13) and beyond from Andrew Howroyd, Aug 19 2018

cp's OEIS Frontend

A277996 Number of free pure symmetric multifunctions (with empty expressions allowed) with one atom and n positions.

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PARI

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A317658 Number of positions in the n-th free pure symmetric multifunction (with empty expressions allowed) with one atom.

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A317875 Number of achiral free pure multifunctions with n unlabeled leaves.

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A317654 Number of free pure symmetric multifunctions whose leaves are a strongly normal multiset of size n.

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A317652 Number of free pure symmetric multifunctions whose leaves are an integer partition of n.

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A317653 Number of free pure symmetric multifunctions whose leaves are a normal multiset of size n.

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A317876 Number of free pure symmetric identity multifunctions (with empty expressions allowed) with one atom and n positions.

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