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.

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A323955 Regular triangle read by rows where T(n, k) is the number of set partitions of {1, ..., n} with no block containing k cyclically successive vertices, n >= 1, 2 <= k <= n + 1.

Original entry on oeis.org

1, 1, 2, 1, 4, 5, 4, 10, 14, 15, 11, 36, 46, 51, 52, 41, 145, 184, 196, 202, 203, 162, 631, 806, 855, 869, 876, 877, 715, 3015, 3847, 4059, 4115, 4131, 4139, 4140, 3425, 15563, 19805, 20813, 21056, 21119, 21137, 21146, 21147, 17722, 86144, 109339, 114469
Offset: 1

Views

Author

Gus Wiseman, Feb 10 2019

Keywords

Comments

Cyclically successive means 1 is a successor of n.

Examples

			Triangle begins:
    1
    1    2
    1    4    5
    4   10   14   15
   11   36   46   51   52
   41  145  184  196  202  203
  162  631  806  855  869  876  877
  715 3015 3847 4059 4115 4131 4139 4140
Row 4 counts the following partitions:
  {{13}{24}}      {{12}{34}}      {{1}{234}}      {{1234}}
  {{1}{24}{3}}    {{13}{24}}      {{12}{34}}      {{1}{234}}
  {{13}{2}{4}}    {{14}{23}}      {{123}{4}}      {{12}{34}}
  {{1}{2}{3}{4}}  {{1}{2}{34}}    {{124}{3}}      {{123}{4}}
                  {{1}{23}{4}}    {{13}{24}}      {{124}{3}}
                  {{12}{3}{4}}    {{134}{2}}      {{13}{24}}
                  {{1}{24}{3}}    {{14}{23}}      {{134}{2}}
                  {{13}{2}{4}}    {{1}{2}{34}}    {{14}{23}}
                  {{14}{2}{3}}    {{1}{23}{4}}    {{1}{2}{34}}
                  {{1}{2}{3}{4}}  {{12}{3}{4}}    {{1}{23}{4}}
                                  {{1}{24}{3}}    {{12}{3}{4}}
                                  {{13}{2}{4}}    {{1}{24}{3}}
                                  {{14}{2}{3}}    {{13}{2}{4}}
                                  {{1}{2}{3}{4}}  {{14}{2}{3}}
                                                  {{1}{2}{3}{4}}

Crossrefs

First column (k = 2) is A000296. Second column (k = 3) is A323949. Rightmost terms are A000110. Second to rightmost terms are A058692.
Cf. A000126, A000325, A001610, A001644, A169985, A306351, A306357, A323950, A323954.

Programs

  • Mathematica
    spsu[,{}]:={{}};spsu[foo,set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,_}];
Table[Length[spsu[Select[Subsets[Range[n]],Select[Partition[Range[n],k,1,1],Function[ed,UnsameQ@@ed&&Complement[ed,#]=={}]]=={}&],Range[n]]],{n,7},{k,2,n+1}]

A196534 Number of different ways to select disjoint nonempty subsets from {1..n} with equal element sum.

Original entry on oeis.org

1, 3, 8, 18, 39, 83, 179, 388, 857, 1914, 4494, 10844, 26923, 70645, 192297, 538646, 1579602, 4793718, 15010425, 48941642, 164010913, 566065123, 2025354291, 7450901462, 27986863322, 107940691328
Offset: 1

Views

Author

Alois P. Heinz, Oct 03 2011

Keywords

Comments

A000225(n) <= a(n) <= A058692(n+1).

Examples

			a(3) = 8: {{1}}, {{2}}, {{3}}, {{1,2}}, {{1,3}}, {{2,3}}, {{1,2,3}}, {{1,2},{3}}. Element sums are 1, 2, 3, 3, 4, 5, 6, and 3, respectively.

Crossrefs

Row sums of A196231. Cf. A000225, A161943, A164934, A164949, A196232, A196233, A196234, A196235, A196236, A196237, A058692.

Programs

  • Maple
    b:= proc(l, n, k) option remember; local i, j; `if`(l=[0$k], 1, `if`(add(j, j=l)>n*(n-1)/2, 0, b(l, n-1, k))+ add(`if`(l[j]-n<0, 0, b(sort([seq(l[i] -`if`(i=j, n, 0), i=1..k)]), n-1, k)), j=1..k)) end: a:= n-> add(add(b([t$k], n, k), t=2*k-1..floor(n*(n+1)/(2*k)))/k!, k=1..n): seq(a(n), n=1..15);
  • Mathematica
    b[l_, n_, k_] := b[l, n, k] = If[l == Array[0&, k], 1, If[Total[l] > n*(n-1)/2, 0, b[l, n-1, k]] + Sum[If[l[[j]]-n < 0, 0, b[Sort[Table[ l[[i]] - If[i == j, n, 0], {i, 1, k}]], n-1, k]], {j, 1, k}]];
a[n_] := Sum[Sum[b[Array[t&, k], n, k], {t, 2*k-1, Floor[n*(n+1)/(2*k)]} ]/k!, {k, 1, Ceiling[n/2]}];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 25}] (* Jean-François Alcover, Jun 01 2022, after Alois P. Heinz *)

Extensions

a(26) from Alois P. Heinz, Oct 20 2014

A347340 E.g.f.: exp( exp(exp(x) - 1) - exp(x) ).

Original entry on oeis.org

1, 0, 1, 4, 17, 91, 587, 4327, 35604, 323316, 3210600, 34574453, 400893066, 4975247460, 65755573847, 921535225267, 13643496840808, 212688569520955, 3480978391442106, 59657975022473437, 1068151956803180295, 19937983367649562025, 387243759600707804811, 7812456801157894913964
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 27 2021

Keywords

Comments

Exponential transform of A058692.
Stirling transform of A000296.

Links

Crossrefs

Cf. A000110, A000166, A000258, A000296, A000587, A052852, A058692, A182386, A288268.

Programs

  • Maple
    g:= proc(n) option remember; `if`(n=0, 1,
      add(g(n-j)*binomial(n-1, j-1), j=2..n))
    end:
b:= proc(n, m) option remember; `if`(n=0,
      g(m), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..23);  # Alois P. Heinz, Aug 27 2021
# second Maple program:
b:= proc(n, t) option remember; `if`(n=0, 1, add(b(n-j, t)*
      `if`(t=0, 1, b(j, 0)-1)*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..23);  # Alois P. Heinz, Sep 02 2021
  • Mathematica
    nmax = 23; CoefficientList[Series[Exp[Exp[Exp[x] - 1] - Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] (BellB[k] - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 23}]
  • PARI
    my(x='x+O('x^25)); Vec(serlaplace(exp(exp(exp(x)-1)-exp(x)))) \\ Michel Marcus, Aug 27 2021

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * (Bell(k) - 1) * a(n-k).
a(n) = Sum_{k=0..n} Stirling2(n,k) * A000296(k).
a(n) = Sum_{k=0..n} binomial(n,k) * A000258(k) * A000587(n-k).
Previous Showing 11-13 of 13 results.

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