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 1-5 of 5 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)

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

Views

Author

Gus Wiseman, Aug 09 2018

Keywords

Comments

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.

Examples

			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]

Links

Crossrefs

Cf. A002033, A003238, A052893, A053492, A214577, A277996, A280000, A317853, A317875.
Cf. A317882, A317884, A317885.

Programs

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

Formula

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

Extensions

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

Views

Author

Gus Wiseman, Aug 09 2018

Keywords

Comments

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.

Examples

			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[][][][][]

Links

Crossrefs

Cf. A002033, A003238, A052893, A053492, A214577, A277996, A280000, A317853, A317875.
Cf. A317882, A317883, A317885.

Programs

  • Maple
    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
  • Mathematica
    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 *)
  • PARI
    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
  • PARI
    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

Formula

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).

Extensions

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

Views

Author

Gus Wiseman, Aug 09 2018

Keywords

Comments

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.

Examples

			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]

Links

Crossrefs

Cf. A001003, A001678, A002033, A003238, A052893, A053492, A067824, A167865, A214577, A317853, A317875.
Cf. A317882, A317883, A317884.

Programs

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

Formula

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

Extensions

Terms a(17) and beyond from Andrew Howroyd, Aug 19 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 1-5 of 5 results.

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