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.

A007837 Number of partitions of n-set with distinct block sizes.

Original entry on oeis.org

1, 1, 1, 4, 5, 16, 82, 169, 541, 2272, 17966, 44419, 201830, 802751, 4897453, 52275409, 166257661, 840363296, 4321172134, 24358246735, 183351656650, 2762567051857, 10112898715063, 62269802986835, 343651382271526, 2352104168848091, 15649414071734847
Offset: 0

Views

Author

Arnold Knopfmacher

Keywords

Comments

Conjecture: the Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and positive integers n and k. Cf. A185895. - Peter Bala, Mar 17 2022

Examples

			From _Gus Wiseman_, Jul 13 2019: (Start)
The a(1) = 1 through a(5) = 16 set partitions with distinct block sizes:
  {{1}}  {{1,2}}  {{1,2,3}}    {{1,2,3,4}}    {{1,2,3,4,5}}
                  {{1},{2,3}}  {{1},{2,3,4}}  {{1},{2,3,4,5}}
                  {{1,2},{3}}  {{1,2,3},{4}}  {{1,2},{3,4,5}}
                  {{1,3},{2}}  {{1,2,4},{3}}  {{1,2,3},{4,5}}
                               {{1,3,4},{2}}  {{1,2,3,4},{5}}
                                              {{1,2,3,5},{4}}
                                              {{1,2,4},{3,5}}
                                              {{1,2,4,5},{3}}
                                              {{1,2,5},{3,4}}
                                              {{1,3},{2,4,5}}
                                              {{1,3,4},{2,5}}
                                              {{1,3,4,5},{2}}
                                              {{1,3,5},{2,4}}
                                              {{1,4},{2,3,5}}
                                              {{1,4,5},{2,3}}
                                              {{1,5},{2,3,4}}
(End)

Links

Crossrefs

Row sums of A131632 or A262072 or A262078 or A309992.
Cf. A000110, A005651, A007838, A032011, A035470, A038041, A178682, A265950, A271423, A275780, A326026, A326514, A326517, A326533.
Column k=0 of A327869.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(add((-d)*(-d!)^(-k/d),
       d=numtheory[divisors](k))*(n-1)!/(n-k)!*a(n-k), k=1..n))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 06 2008
# second Maple program:
A007837 := proc(n) option remember; local k; `if`(n = 0, 1,
add(binomial(n-1, k-1) * A182927(k) * A007837(n-k), k = 1..n)) end:
seq(A007837(i),i=0..24); # Peter Luschny, Apr 25 2011
  • Mathematica
    nn=20;p=Product[1+x^i/i!,{i,1,nn}];Drop[Range[0,nn]!CoefficientList[ Series[p,{x,0,nn}],x],1]  (* Geoffrey Critzer, Sep 22 2012 *)
a[0]=1; a[n_] := a[n] = Sum[(n-1)!/(n-k)!*DivisorSum[k, -#*(-#!)^(-k/#)&]* a[n-k], {k, 1, n}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 23 2015, after Vladeta Jovovic *)
  • PARI
    {my(n=20); Vec(serlaplace(prod(k=1, n, (1+x^k/k!) + O(x*x^n))))} \\ Andrew Howroyd, Dec 21 2017

Formula

E.g.f.: Product_{m >= 1} (1+x^m/m!).
a(n) = Sum_{k=1..n} (n-1)!/(n-k)!*b(k)*a(n-k), where b(k) = Sum_{d divides k} (-d)*(-d!)^(-k/d) and a(0) = 1. - Vladeta Jovovic, Oct 13 2002
E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*x^(j*k)/(k*(j!)^k)). - Ilya Gutkovskiy, Jun 18 2018

Extensions

More terms from Christian G. Bower
a(0)=1 prepended by Alois P. Heinz, Aug 29 2015

A262072 Number T(n,k) of partitions of an n-set with distinct block sizes and maximal block size equal to k; triangle T(n,k), n>=0, ceiling((sqrt(1+8*n)-1)/2)<=k<=n, read by rows.

Original entry on oeis.org

1, 1, 1, 3, 1, 4, 1, 10, 5, 1, 60, 15, 6, 1, 140, 21, 7, 1, 280, 224, 28, 8, 1, 1260, 630, 336, 36, 9, 1, 12600, 3780, 1050, 480, 45, 10, 1, 34650, 7392, 1650, 660, 55, 11, 1, 110880, 74844, 12672, 2475, 880, 66, 12, 1, 360360, 276276, 140712, 20592, 3575, 1144, 78, 13, 1
Offset: 0

Views

Author

Alois P. Heinz, Sep 10 2015

Keywords

Examples

			Triangle T(n,k) begins:
: 1;
:    1;
:       1;
:       3,  1;
:           4,     1;
:          10,     5,    1;
:          60,    15,    6,    1;
:                140,   21,    7,   1;
:                280,  224,   28,   8,  1;
:               1260,  630,  336,  36,  9,  1;
:              12600, 3780, 1050, 480, 45, 10, 1;

Links

Crossrefs

Row sums give A007837.
Column sums give A262073.
Cf. A002024, A262071, A262078 (same read by columns).

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(i*(i+1)/2n, 0, binomial(n, i)*b(n-i, i-1))))
    end:
T:= (n, k)-> b(n, k) -`if`(k=0, 0, b(n, k-1)):
seq(seq(T(n,k), k=ceil((sqrt(1+8*n)-1)/2)..n), n=0..14);
  • Mathematica
    b[n_, i_] := b[n, i] = If[i*(i+1)/2n, 0, Binomial[n, i]*b[n-i, i-1]]]]; T[n_, k_] := b[n, k] - If[k == 0, 0, b[n, k-1]]; Table[T[n, k], {n, 0, 14}, {k, Ceiling[(Sqrt[1+8*n]-1)/2], n}] // Flatten (* Jean-François Alcover, Feb 04 2017, translated from Maple *)

A262073 Number of partitions of k-sets with distinct block sizes and maximal block size equal to n (n <= k <= n*(n+1)/2).

Original entry on oeis.org

1, 1, 4, 75, 14301, 40870872, 2163410250576, 2525542278491543715, 75742007488274337351844747, 66712890687959224726994385259183993, 1942822997098466460791474215498474580001684381, 2080073366817374333366496031890682227244159986035768679984
Offset: 0

Views

Author

Alois P. Heinz, Sep 10 2015

Keywords

Comments

a(n)^(1/n^2) / sqrt(n) tends to exp(1/4)/sqrt(2) = 0.907943... . - Vaclav Kotesovec, May 14 2016

Links

Crossrefs

Column sums of A262072 or A262078.
Cf. A000217.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(i*(i+1)/2n, 0, binomial(n, i)*b(n-i, i-1))))
    end:
T:= (n, k)-> b(n, k) -`if`(k=0, 0, b(n, k-1)):
a:= n-> add(T(k, n), k=n..n*(n+1)/2):
seq(a(n), n=0..14);
  • Mathematica
    b[n_, i_] := b[n, i] = If[i*(i + 1)/2 < n, 0, If[n == 0, 1, b[n, i - 1] + If[i > n, 0, Binomial[n, i]*b[n - i, i - 1]]]];
T[n_, k_] := b[n, k] - If[k == 0, 0, b[n, k - 1]];
a[n_] := Sum[T[k, n], { k, n, n*(n + 1)/2}];
Table[a[n], {n, 0, 14}] (* Jean-François Alcover, May 24 2018, translated from Maple *)

Formula

a(n) = Sum_{k=n..n*(n+1)/2} A262072(k,n).
Showing 1-3 of 3 results.

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