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 1-4 of 4 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

A325901 Numbers having at least two representations as multinomial coefficients M(n;lambda), where lambda is a partition of n into distinct parts.

Original entry on oeis.org

1, 10, 15, 21, 28, 35, 36, 45, 55, 56, 60, 66, 78, 84, 91, 105, 120, 126, 136, 153, 165, 168, 171, 190, 210, 220, 231, 252, 253, 276, 280, 286, 300, 325, 330, 351, 360, 364, 378, 406, 435, 455, 462, 465, 495, 496, 504, 528, 560, 561, 595, 630, 660, 666, 680
Offset: 1

Views

Author

Alois P. Heinz, Sep 07 2019

Keywords

Comments

Numbers that are repeated in the triangle A309992 (all positive integers except 2 occur at least once).
All triangular numbers (A000217) except 0, 3 and 6 are in this sequence.
All terms are also contained in A325472.

Examples

			1 is in the sequence because M(0;0) = M(1;1) = M(2;2) = M(3;3) = ... = 1.
10 is in the sequence because M(10;9,1) = M(5;3,2) = 10.
55 is in the sequence because M(55;54,1) = M(11;9,2) = 55.
105 is in the sequence because M(7;4,2,1) = M(15;13,2) = M(105;104,1) = 105.

Links

Crossrefs

Cf. A000009, A000217, A309992, A325472, A325903.

A325903 Numbers having at least three representations as multinomial coefficients M(n;lambda), where lambda is a partition of n into distinct parts.

Original entry on oeis.org

1, 105, 120, 210, 495, 1260, 1365, 1540, 3003, 4620, 5460, 6435, 7140, 10296, 11628, 15504, 24310, 27720, 29260, 30030, 42504, 43680, 45045, 77520, 83160, 102960, 116280, 120120, 180180, 203490, 352716, 360360, 376740, 437580, 593775, 657800, 680680, 720720
Offset: 1

Views

Author

Alois P. Heinz, Sep 07 2019

Keywords

Comments

Numbers occurring at least three times in the triangle A309992.
All terms are contained in A325593 and in A325901.

Examples

			1 is in the sequence because M(0;0) = M(1;1) = M(2;2) = M(3;3) = ... = 1.
105 is in the sequence because M(7;4,2,1) = M(15;13,2) = M(105;104,1) = 105.
120 is in the sequence because M(10;7,3) = M(16;14,2) = M(120;119,1) = 120.
1365 is in the sequence because M(15;11,4) = M(15;12,2,1) = M(1365;1364,1) = 1365.

Links

Crossrefs

Cf. A000009, A309992, A325593, A325901.

A290517 Maximum value of the multinomial coefficients M(n;lambda), where lambda ranges over all partitions of n into distinct parts.

Original entry on oeis.org

1, 1, 1, 3, 4, 10, 60, 105, 280, 1260, 12600, 27720, 83160, 360360, 2522520, 37837800, 100900800, 343062720, 1543782240, 9777287520, 97772875200, 2053230379200, 6453009763200, 24736537425600, 118735379642880, 742096122768000, 6431499730656000
Offset: 0

Views

Author

Alois P. Heinz, Aug 04 2017

Keywords

Examples

			a(10) = 12600 = 10! / (4! * 3! * 2! * 1!) is the value for partition [4,3,2,1]. All other partitions of 10 into distinct parts give smaller values: [5,3,2]-> 2520, [5,4,1]-> 1260, [6,3,1]-> 840, [6,4]-> 210, [7,2,1]-> 360, [7,3]-> 120, [8,2]-> 45, [9,1]-> 10, [10]-> 1.

Links

Crossrefs

Cf. A000009, A000142, A000178, A004736, A022915, A290518, A309992.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n>i*(i+1)/2, infinity,
     `if`(n=0, 1, min(b(n, i-1), b(n-i, min(n-i, i-1))*i!)))
    end:
a:= n-> n!/b(n$2):
seq(a(n), n=0..30);
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, a(n-1)*n/
      (t-> t*(t+3)/2-n+2)(floor(sqrt(8*n-7)/2-1/2)))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Aug 05 2017
  • Mathematica
    b[n_, i_]:=b[n, i]=If[n>i*(i + 1)/2, Infinity, If[n==0, 1, Min[b[n, i - 1], b[n - i, Min[n - i, i - 1]]*i!]]]; Table[n!/b[n, n], {n, 0, 30}] (* Indranil Ghosh, Aug 05 2017, after Maple *)

Formula

a(n) = A000142(n) / A290518(n).
a(0) = 1, a(n) = n * a(n-1) / A004736(n) for n>0.
a(n) = A309992(n,A000009(n)). - Alois P. Heinz, Aug 26 2019
Showing 1-4 of 4 results.

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