A317875 -id:A317875 - cp's OEIS frontend

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

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

A317879 Number of free pure identity multifunctions (with empty expressions allowed) with one atom and n positions.

Original entry on oeis.org

1, 1, 2, 4, 11, 29, 83, 251, 767, 2403, 7652, 24758, 80875, 266803, 887330, 2972108, 10016981, 33942461, 115572864, 395226810, 1356840007, 4674552089, 16156355357, 56003840659, 194651585875, 678220460687, 2368505647624, 8288873657180, 29064904732911

Offset: 1

Gus Wiseman, Aug 09 2018

nonn

A free pure identity multifunction (with empty expressions allowed) (IME) 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 IME, each of the g_i for i = 1, ..., k >= 0 is an IME, and for i != j we have g_i != g_j. The number of positions in an IME is the number of brackets [...] plus the number of o's.

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

			The a(5) = 11 IMEs:
  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..500

Cf. A000081, A001003, A004111, A277996, A280000, A317875.

Cf. A317876, A317877, A317878, 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[Tuples[allIdExpr/@p],UnsameQ@@#&]}],{p,Join@@Permutations/@IntegerPartitions[g]}]]];
Table[Length[allIdExpr[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]=v[n-1]+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

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

A317881 Number of series-reduced free pure identity multifunctions (with empty expressions allowed) with one atom and n positions.

Original entry on oeis.org

1, 1, 1, 1, 3, 7, 15, 37, 91, 231, 593, 1557, 4111, 10941, 29295, 79087, 215015, 587463, 1611985, 4441473, 12284513, 34095797, 94931525, 265061363, 742029431, 2082310665, 5856540305, 16505796865, 46608877763, 131850193107, 373612733107, 1060339387939, 3013758348317

Offset: 1

Gus Wiseman, Aug 09 2018

nonn

A series-reduced series-reduced free pure identity multifunction (with empty expressions allowed) (SRIM) 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 SRIM, k is an integer greater than or equal to 0 but not equal to 1, each of the g_i for i = 1, ..., k >= 0 is an SRIM, and for i != j we have g_i != g_j. The number of positions in an SRIM is the number of brackets [...] plus the number of o's.

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

			The a(6) = 7 SRIMs:
  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, A067824, A167865, A277996, A280000, A317875.

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

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[Tuples[allIdExprSR/@p],UnsameQ@@#&]}],{p,If[g==0,{{}},Join@@Permutations/@Rest[IntegerPartitions[g]]]}]]];
Table[Length[allIdExprSR[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]=v[n-1]+sum(k=1, n-2, v[n-k-1]*(subst(serlaplace(y^0*polcoef(p, k)), y, 1)-v[k]))); v} \\ Andrew Howroyd, Sep 01 2018

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

A317882 Number of free pure achiral multifunctions (with empty expressions allowed) with one atom and n positions.

Original entry on oeis.org

1, 1, 2, 5, 12, 31, 79, 211, 564, 1543, 4259, 11899, 33526, 95272, 272544, 784598, 2270888, 6604900, 19293793, 56581857, 166523462, 491674696, 1455996925, 4323328548, 12869353254, 38396655023, 114803257039, 343932660450, 1032266513328, 3103532577722

Offset: 1

Gus Wiseman, Aug 09 2018

nonn

A free pure achiral multifunction (with empty expressions allowed) (AME) is either (case 1) the leaf symbol "o", or (case 2) a possibly empty expression of the form h[g, ..., g] where h and g are AMEs. The number of positions in an AME is the number of brackets [...] plus the number of o's.

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

			The a(5) = 12 AMEs:
  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, A002033, A003238, A052893, A053492, A214577, A277996, A317853, A317875.

Cf. A317883, A317884, A317885.

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

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

a(1) = 1; a(n > 1) = a(n - 1) + Sum_{0 < k < n - 1} a(k) * Sum_{d|(n - k - 1)} a(d).

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

A317883 Number of free pure achiral multifunctions with one atom and n positions.

Original entry on oeis.org

1, 0, 1, 1, 3, 4, 10, 17, 37, 70, 150, 299, 634, 1311, 2786, 5879, 12584, 26904, 58005, 125242, 271819, 591297, 1290976, 2825170, 6199964, 13635749, 30057649, 66386206, 146903289, 325637240, 723024160, 1607805207, 3580476340, 7984266625, 17827226469

Offset: 1

Gus Wiseman, Aug 09 2018

nonn

A free pure achiral multifunction (PAM) 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 PAMs. The number of positions in a PAM is the number of brackets [...] plus the number of o's.

			The a(7) = 10 PAMs:
  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. A002033, A003238, A052893, A053492, A214577, A277996, A280000, A317853, A317875.

Cf. A317882, A317884, A317885.

a[n_]:=If[n==1,1,Sum[a[k]*Sum[a[d],{d,Divisors[n-k-1]}],{k,n-2}]];
Array[a,12]

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

a(1) = 1; a(n > 1) = Sum_{0 < k < n - 1} a(k) * Sum_{d|(n - k - 1)} a(d).

G.f. A(x) satisfies: A(x) = x * (1 + A(x) * Sum_{k>=1} A(x^k)). - Ilya Gutkovskiy, May 03 2019

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

A317884 Number of series-reduced achiral free pure multifunctions (with empty expressions allowed) with one atom and n positions.

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 14, 26, 47, 87, 160, 295, 540, 997, 1832, 3369, 6197, 11406, 20975, 38594, 70991, 130610, 240275, 442043, 813184, 1496053, 2752251, 5063319, 9314879, 17136632, 31526032, 57998423, 106699160, 196294065, 361120800, 664352454, 1222204958

Offset: 1

Gus Wiseman, Aug 09 2018

nonn

A series-reduced achiral expression (SRAE) is either (case 1) the leaf symbol "o", or (case 2) a possibly empty but not unitary expression of the form h[g, ..., g], where h and g are SRAEs. The number of positions in an SRAE is the number of brackets [...] plus the number of o's.

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

			The a(6) = 8 SRAEs:
  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..500

Cf. A002033, A003238, A052893, A053492, A214577, A277996, A280000, A317853, A317875.

Cf. A317882, A317883, A317885.

a:= proc(n) option remember; `if`(n=1, 1, a(n-1)+add(a(j)*add(
      a(d), d=numtheory[divisors](n-j-1) minus {n-j-1}), j=1..n-1))
    end:
seq(a(n), n=1..45);  # Alois P. Heinz, Sep 05 2018

allAchExprSR[n_] := If[n == 1, {"o"}, Join @@ Cases[Table[PR[k, n - k - 1], {k, n - 1}], PR[h_, g_] :> Join @@ Table[Apply @@@ Tuples[{allAchExprSR[h], Select[Tuples[allAchExprSR /@ p], SameQ @@ # &]}], {p, If[g == 0, {{}}, Join @@ Permutations /@ Rest[IntegerPartitions[g]]]}]]]; Table[Length[allAchExprSR[n]], {n, 12}]
(* Second program: *)
a[n_] := a[n] = If[n == 1, 1, a[n-1] + Sum[a[j]*DivisorSum[
     n-j-1, If[# < n-j-1, a[#], 0]&], {j, 1, n-2}]];
Array[a, 45] (* Jean-François Alcover, May 17 2021, after Alois P. Heinz *)

seq(n)={my(p=O(x)); for(n=1, n, p = x + p*x*(1 + sum(k=2, n-2, 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]=v[n-1] + sum(i=1, n-2, v[i]*sumdiv(n-i-1, d, if(dAndrew Howroyd, Aug 19 2018

a(1) = 1; a(n > 1) = a(n-1) + Sum_{0 < k < n-1} a(k) * Sum_{d|(n-k-1), d < n-k-1} a(d).

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

A317885 Number of series-reduced free pure achiral multifunctions with one atom and n positions.

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 2, 3, 4, 7, 9, 14, 21, 32, 45, 69, 103, 153, 224, 338, 500, 746, 1107, 1645, 2447, 3652, 5413, 8052, 11993, 17834, 26500, 39447, 58655, 87240, 129772, 193001, 287034, 427014, 635048, 944501, 1404910, 2089633, 3107864, 4622670, 6875533

Offset: 1

Gus Wiseman, Aug 09 2018

nonn

A series-reduced free pure achiral multifunction (SRAM) is either (case 1) the leaf symbol "o", or (case 2) a nonempty and non-unitary expression of the form h[g, ..., g] where h and g are SRAMs. The number of positions in a SRAM is the number of brackets [...] plus the number of o's.

			The a(10) = 7 SRAMs:
  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,o,o,o,o,o]

Andrew Howroyd, Table of n, a(n) for n = 1..500

Cf. A001003, A001678, A002033, A003238, A052893, A053492, A067824, A167865, A214577, A317853, A317875.

Cf. A317882, A317883, A317884.

a[n_]:=If[n==1,1,Sum[a[k]*Sum[a[d],{d,Most[Divisors[n-k-1]]}],{k,n-2}]];
Array[a,12]

seq(n)={my(p=O(x)); for(n=1, n, p = x + p*x*sum(k=2, n-2, 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-2, v[i]*sumdiv(n-i-1, d, if(dAndrew Howroyd, Aug 19 2018

a(1) = 1; a(n > 1) = Sum_{0 < k < n - 1} a(k) * Sum_{d|(n - k - 1), d < n - k - 1} a(d).

Terms a(17) and beyond from Andrew Howroyd, Aug 19 2018

cp's OEIS Frontend

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

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

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

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A317879 Number of free pure identity multifunctions (with empty expressions allowed) with one atom and n positions.

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

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A317881 Number of series-reduced free pure identity multifunctions (with empty expressions allowed) with one atom and n positions.

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A317882 Number of free pure achiral multifunctions (with empty expressions allowed) with one atom and n positions.

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