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 1-3 of 3 results.

A095989 INVERTi transform applied to the ordered Bell numbers.

Original entry on oeis.org

1, 2, 8, 48, 368, 3376, 35824, 430512, 5773936, 85482032, 1384936688, 24380214960, 463522810736, 9468048895792, 206831329017328, 4812581925690288, 118843801816575088, 3104590192664327216, 85544737118902122224, 2479681575659312797872, 75434373300016828382576
Offset: 1

Views

Author

Mike Zabrocki, Jul 18 2004

Keywords

Comments

A set composition of n is an ordered sequence [S_1, S_2, ..., S_k] where S_i subset of [n] all disjoint and the union of all S_i is [n] (see A000670). A set composition is atomic if S_1 union ... union S_j does not equal [r] for any r < n and j < k. a(n) is the number of atomic set compositions.
A preference function of n is a word of length n where all the numbers 1 through k occur at least once for some k <= n (see A000670). A preference function is atomic if no strict leading subword contains the only occurrences in the word of the letters 1 through j < k. a(n) is the number of atomic preference functions.

Examples

			Atomic set compositions a(1)=1: [{1}]; a(2)=2: [{12}], [{2},{1}]; a(3)=8: [{123}], [{2},{13}], [{3}, {12}], [{23}, {1}], [{13},{2}], [{2},{3},{1}], [{3},{1},{2}], [{3},{2},{1}].
Atomic preference functions a(1) = 1: 1; a(2)=2: 11, 21; a(3)=8: 111, 212, 221, 211, 121, 312, 231, 321.

Links

Crossrefs

Cf. A000670, A074664, A095993.

Programs

  • Maple
    A000670 := proc(n) option remember; local k; if n <=1 then 1 else add(binomial(n,k)*A000670(n-k),k=1..n); fi; end: add(A000670(k)*x^k,k=0..20): series(1-1/%,x,21): [seq(coeff(%,x,i),i=1..20)];
  • Mathematica
    max = 20; Fubini[n_, r_] := Sum[k!*Sum[(-1)^(i+k+r)*(i+r)^(n-r)/(i!*(k-i-r)!), {i, 0, k-r}], {k, r, n}]; Fubini[0, 1] = 1; s = 1 - 1/Sum[ Fubini[k, 1] q^k, {k, 0, max}] + O[q]^max; CoefficientList[s/q, q] (* Jean-François Alcover, Mar 31 2016 *)

Formula

G.f.: 1 - 1/Sum_{k>=0} A000670(k)*q^k.
G.f.: x/(1-2x/(1-2x/(1-4x/(1-3x/(1-6x/(1-4x/(1-8x/(1-5x/(1-...(continued fraction). - Philippe Deléham, Nov 22 2011
G.f.: (1-T(0))/x, where T(k) = 1 - x*(k+1)/(1 - 2*x*(k+1)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 29 2013
Let A(x) be the g.f. A095989, B(x) the g.f. A000670, then A(x) = (1 - 1/B(x))/x. - Sergei N. Gladkovskii, Nov 29 2013
a(n) ~ n! / (2 * log(2)^(n+1)). - Vaclav Kotesovec, Oct 09 2019

A290352 Euler transform of the Fubini numbers (ordered Bell numbers, A000670).

Original entry on oeis.org

1, 1, 4, 17, 98, 678, 5687, 55656, 626161, 7963511, 113027113, 1770785023, 30346490633, 564546034917, 11327726548719, 243811768229012, 5602495216123312, 136878883607160468, 3542830077444873188, 96835203745704714722, 2787051847418347608600
Offset: 0

Views

Author

Alois P. Heinz, Jul 28 2017

Keywords

Links

Crossrefs

Cf. A000670, A007003, A095993, A290351.

Programs

  • Maple
    b:= proc(n, m) option remember;
     `if`(n=0, m!, m*b(n-1, m)+b(n-1, m+1))
    end:
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      b(d, 0), d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..30);
  • Mathematica
    b[n_, m_]:=b[n, m]=If[n==0, m!, Sum[b[n - 1, Max[m, j]], {j, m + 1}]]; a[n_]:=a[n]=If[n==0, 1, Sum[Sum[d*b[d, 0], {d, Divisors[j]}] a[n - j], {j, n}]/n]; Table[a[n], {n, 0, 50}] (* Indranil Ghosh, Jul 28 2017, after Maple code *)

Formula

a(n) ~ n! / (2 * (log(2))^(n+1)). - Vaclav Kotesovec, May 31 2019

A305853 Inverse Weigh transform of the Fubini numbers (ordered Bell numbers, A000670).

Original entry on oeis.org

1, 3, 10, 62, 446, 3975, 41098, 484152, 6390488, 93419965, 1498268466, 26159940522, 494036061550, 10035451747919, 218207845446062, 5057251219752612, 124462048466812950, 3241773988594489244, 89093816361187396674, 2576652694087236419386, 78224564280680539732266
Offset: 1

Views

Author

Alois P. Heinz, Jun 11 2018

Keywords

Links

Crossrefs

Cf. A000670, A095993, A305846, A305852.

Programs

  • Maple
    g:= proc(n) option remember; `if`(n=0, 1,
      add(g(n-j)*binomial(n, j), j=1..n))
    end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(a(i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= proc(n) option remember; g(n)-b(n, n-1) end:
seq(a(n), n=1..30);
  • Mathematica
    g[n_] := g[n] = If[n == 0, 1,
    Sum[g[n - j] Binomial[n, j], {j, 1, n}]];
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0,
    Sum[Binomial[a[i], j] b[n - i j, i - 1], {j, 0, n/i}]]];
a[n_] := a[n] = g[n] - b[n, n - 1];
a /@ Range[1, 30] (* Jean-François Alcover, Dec 21 2020, after Alois P. Heinz *)

Formula

Product_{k>=1} (1+x^k)^a(k) = Sum_{n>=0} A000670(n) * x^n.
a(n) ~ n! / (2 * log(2)^(n+1)). - Vaclav Kotesovec, Sep 10 2019
Showing 1-3 of 3 results.

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