cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

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

Views

Author

Gus Wiseman, Aug 09 2018

Keywords

Comments

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.

Examples

			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].

Links

Crossrefs

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.

Programs

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

Formula

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)

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

Views

Author

Gus Wiseman, Aug 09 2018

Keywords

Comments

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.

Examples

			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[][][][]

Links

Crossrefs

Cf. A000081, A004111, A052893, A053492, A277996, A280000, A317652, A317653, A317654, A317875.
Cf. A317877, A317878, A317879, A317880, A317881.

Programs

  • Mathematica
    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}]
  • PARI
    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

Formula

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)

Extensions

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

Views

Author

Gus Wiseman, Aug 09 2018

Keywords

Comments

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.

Examples

			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]

Links

Crossrefs

Cf. A000081, A001003, A003238, A004111, A052893, A053492, A277996, A280000, A317875.
Cf. A317876, A317878, A317879, A317880, A317881.

Programs

  • Mathematica
    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}]
  • PARI
    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

Extensions

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

Views

Author

Gus Wiseman, Aug 09 2018

Keywords

Comments

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.

Examples

			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]

Links

Crossrefs

Cf. A000081, A003238, A004111, A052893, A053492, A277996, A280000, A317875.
Cf. A317876, A317877, A317879, A317880, A317881.

Programs

  • Mathematica
    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}]
  • PARI
    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

Extensions

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

Views

Author

Gus Wiseman, Aug 09 2018

Keywords

Comments

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.

Examples

			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[][][][][][]

Links

Crossrefs

Cf. A001003, A001678, A052893, A053492, A067824, A167865, A277996, A280000, A317875.
Cf. A317876, A317877, A317878, A317879, A317880, A317881.

Programs

  • Mathematica
    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}]
  • PARI
    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

Extensions

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

Views

Author

Gus Wiseman, Aug 09 2018

Keywords

Comments

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.

Examples

			The a(6) = 7 SRIMs:
  o[o[][],o]
  o[o,o[][]]
  o[][o[],o]
  o[][o,o[]]
  o[o[],o][]
  o[o,o[]][]
  o[][][][][]

Links

Crossrefs

Cf. A001003, A001678, A067824, A167865, A277996, A280000, A317875.
Cf. A317876, A317877, A317878, A317879, A317880.

Programs

  • Mathematica
    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}]
  • PARI
    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

Extensions

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

A317853 a(1) = 1; a(n > 1) = Sum_{0 < k < n} (-1)^(n - k - 1) a(n - k) Sum_{d|k} a(d).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 5, 6, 11, 14, 23, 26, 51, 70, 114, 147, 237, 314, 516, 715, 1118, 1549, 2353, 3252, 5011, 7235, 10724, 15142, 22504, 32506, 47770, 69173, 100980, 146657, 212504, 308563, 448256, 658037, 946166, 1373739, 1988283, 2919185, 4197886, 6118850
Offset: 1

Views

Author

Gus Wiseman, Aug 09 2018

Keywords

Crossrefs

Cf. A001003, A001678, A002033, A003238, A052893, A053492, A067824, A167865, A214577, A277996, A280000, A317875.
Cf. A317876, A317877, A317878, A317879, A317880, A317881.
Cf. A317882, A317883, A317884, A317885.

Programs

  • Mathematica
    a[n_]:=a[n]=If[n==1,1,Sum[(-1)^(n-k-1)*a[n-k]*Sum[a[d],{d,Divisors[k]}],{k,n-1}]];
Array[a,50]
Showing 1-7 of 7 results.

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