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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A115278 Number of partitions of {1,...,2*n} into even sized blocks such that no block size is repeated.

Original entry on oeis.org

1, 1, 1, 16, 29, 256, 14422, 49141, 490429, 10758400, 1797335306, 9458619391, 133756636598, 2528529510391, 137864810180749, 53441183229799381, 410251032050409469, 7615997734377068128, 167055180095977694194, 6741819165851219788075, 738863335901972011745434
Offset: 0

Views

Author

Christian G. Bower, Jan 18 2006

Keywords

Links

Crossrefs

Cf. A005046, A007837, A032310, A115277.

Programs

  • Maple
    with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-2), j=0..min(1, n/i))))
    end:
a:= n-> b(2*n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 08 2015
  • Mathematica
    multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[multinomial[n, Join[{n - i*j}, Array[i &, j]]]/j!* b[n - i*j, i - 2], {j, 0, Min[1, n/i]}]]]; a[n_] := b[2 n, 2 n]; Table[ a[n], {n, 0, 30}] (* Jean-François Alcover, Jan 22 2016, after Alois P. Heinz *)

Formula

E.g.f.: B(x) of b(n) where b(2*n)=a(n), b(2*n+1)=0. B(x)=Product {m >= 1} (1+x^(2*m)/(2*m)!).

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 *)

A275679 Number of set partitions of [n] with alternating block size parities.

Original entry on oeis.org

1, 1, 1, 4, 3, 20, 43, 136, 711, 1606, 12653, 36852, 250673, 1212498, 6016715, 45081688, 196537387, 1789229594, 8963510621, 76863454428, 512264745473, 3744799424978, 32870550965259, 219159966518160, 2257073412153459, 15778075163815474, 165231652982941085
Offset: 0

Views

Author

Alois P. Heinz, Aug 05 2016

Keywords

Examples

			a(3) = 4: 123, 12|3, 13|2, 1|23.
a(4) = 3: 1234, 1|23|4, 1|24|3.
a(5) = 20: 12345, 1234|5, 1235|4, 123|45, 1245|3, 124|35, 125|34, 12|345, 12|3|45, 1345|2, 134|25, 135|24, 13|245, 13|2|45, 145|23, 14|235, 15|234, 1|2345, 14|2|35, 15|2|34.

Links

Crossrefs

Cf. A003724, A005046, A007837, A038041, A275309, A275310, A275311, A275312, A275313, A286076, A361804.

Programs

  • Maple
    b:= proc(n, t) option remember; `if`(n=0, 1, add(
      `if`((i+t)::odd, b(n-i, 1-t)*binomial(n-1, i-1), 0), i=1..n))
    end:
a:= n-> `if`(n=0, 1, b(n, 0)+b(n, 1)):
seq(a(n), n=0..35);
  • Mathematica
    b[n_, t_] := b[n, t] = If[n==0, 1, Sum[If[OddQ[i+t], b[n-i, 1-t] * Binomial[n-1, i-1], 0], {i, 1, n}]]; a[n_] := If[n==0, 1, b[n, 0] + b[n, 1]]; Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Feb 27 2017, translated from Maple *)

A305547 Expansion of e.g.f. Product_{k>=1} (1 + (exp(x) - 1)^k/k!).

Original entry on oeis.org

1, 1, 2, 8, 37, 182, 1039, 7149, 56382, 479220, 4280247, 40406984, 410453366, 4539623168, 54431372233, 695801259947, 9312538336475, 128985882874288, 1842668013046405, 27238267120063415, 419396473955088310, 6769168354222927254, 114837651830425810381, 2042782103293394499566
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 15 2018

Keywords

Comments

Stirling transform of A007837.

Links

Crossrefs

Cf. A007837, A140585, A305987.

Programs

  • Maple
    b:= proc(n) option remember; `if`(n=0, 1, add(add((-d)*(-d!)^(-k/d),
       d=numtheory[divisors](k))*(n-1)!/(n-k)!*b(n-k), k=1..n))
    end:
a:= n-> add(Stirling2(n, k)*b(k), k=0..n):
seq(a(n), n=0..25);  # Alois P. Heinz, Jun 15 2018
  • Mathematica
    nmax = 23; CoefficientList[Series[Product[(1 + (Exp[x] - 1)^k/k!), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 23; CoefficientList[Series[Exp[Sum[Sum[(-1)^(k + 1) (Exp[x] - 1)^(j k)/((j!)^k k), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
b[0] = 1; b[n_] := b[n] = Sum[(n - 1)!/(n - k)! DivisorSum[k, -# (-#!)^(-k/#) &] b[n - k], {k, 1, n}]; a[n_] := a[n] = Sum[StirlingS2[n, k] b[k], {k, 0, n}]; Table[a[n], {n, 0, 23}]

Formula

E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*(exp(x) - 1)^(j*k)/((j!)^k*k)).
a(n) = Sum_{k=0..n} Stirling2(n,k)*A007837(k).

A309992 Triangle T(n,k) whose n-th row lists in increasing order the multinomial coefficients M(n;lambda), where lambda ranges over all partitions of n into distinct parts; n >= 0, 1 <= k <= A000009(n), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 4, 1, 5, 10, 1, 6, 15, 60, 1, 7, 21, 35, 105, 1, 8, 28, 56, 168, 280, 1, 9, 36, 84, 126, 252, 504, 1260, 1, 10, 45, 120, 210, 360, 840, 1260, 2520, 12600, 1, 11, 55, 165, 330, 462, 495, 1320, 2310, 4620, 6930, 27720
Offset: 0

Views

Author

Alois P. Heinz, Aug 26 2019

Keywords

Comments

First row with repeated terms is row 15, see also A309999: 1365 = M(15;11,4) = M(15;12,2,1) and 30030 = M(15;9,5,1) = M(15;10,3,2).

Examples

			For n = 5 there are 3 partitions of 5 into distinct parts: [5], [4,1], [3,2].  So row 5 contains M(5;5) = 1, M(5;4,1) = 5 and M(5;3,2) = 10.
Triangle T(n,k) begins:
  1;
  1;
  1;
  1,  3;
  1,  4;
  1,  5, 10;
  1,  6, 15,  60;
  1,  7, 21,  35, 105;
  1,  8, 28,  56, 168, 280;
  1,  9, 36,  84, 126, 252, 504, 1260;
  1, 10, 45, 120, 210, 360, 840, 1260, 2520, 12600;
  1, 11, 55, 165, 330, 462, 495, 1320, 2310,  4620, 6930, 27720;
  ...

Links

Crossrefs

Columns k=1-3 give: A000012, A000027 (for n>=3), A000217(n-1) (for n>=5).
Row sums give A007837.
Rightmost terms of rows give A290517.
Cf. A000009, A036038, A309999, A325901, A325903.

Programs

  • Maple
    g:= proc(n, i) option remember; `if`(i*(i+1)/2binomial(n, i)*x, g(n-i, min(n-i, i-1)))[], g(n, i-1)[]]))
    end:
T:= n-> sort(g(n$2))[]:
seq(T(n), n=0..14);
  • Mathematica
    g[n_, i_] := g[n, i] = If[i(i+1)/2 < n, {}, If[n == 0, {1}, Join[ Binomial[n, i] # & /@ g[n-i, Min[n-i, i-1]], g[n, i-1]]]];
T[n_] := Sort[g[n, n]];
T /@ Range[0, 14] // Flatten (* Jean-François Alcover, Jan 27 2021, after Alois P. Heinz *)

A361865 Number of set partitions of {1..n} such that the mean of the means of the blocks is an integer.

Original entry on oeis.org

1, 0, 3, 2, 12, 18, 101, 232, 1547, 3768, 24974, 116728, 687419, 3489664, 26436217, 159031250, 1129056772
Offset: 1

Views

Author

Gus Wiseman, Apr 04 2023

Keywords

Examples

			The set partition y = {{1,4},{2,5},{3}} has block-means {5/2,7/2,3}, with mean 3, so y is counted under a(5).
The a(1) = 1 through a(5) = 12 set partitions:
  {{1}}  .  {{123}}      {{1}{234}}  {{12345}}
            {{13}{2}}    {{123}{4}}  {{1245}{3}}
            {{1}{2}{3}}              {{135}{24}}
                                     {{15}{234}}
                                     {{1}{234}{5}}
                                     {{12}{3}{45}}
                                     {{135}{2}{4}}
                                     {{14}{25}{3}}
                                     {{15}{24}{3}}
                                     {{1}{24}{3}{5}}
                                     {{15}{2}{3}{4}}
                                     {{1}{2}{3}{4}{5}}

Crossrefs

For median instead of mean we have A361864.
For sum instead of outer mean we have A361866, median A361911.
A000110 counts set partitions.
A067538 counts partitions with integer mean, ranks A326836, strict A102627.
A308037 counts set partitions whose block-sizes have integer mean.
A327475 counts subsets with integer mean, median A000975.
Cf. A007837, A035470, A038041, A275714, A275780, A326512, A326513, A326521, A326537, A327481.

Programs

  • Mathematica
    sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[Range[n]],IntegerQ[Mean[Mean/@#]]&]],{n,6}]

Extensions

a(13)-a(17) from Christian Sievers, Jun 30 2025

A371788 Triangle read by rows where T(n,k) is the number of set partitions of {1..n} with exactly k distinct block-sums.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 2, 8, 4, 1, 0, 2, 19, 24, 6, 1, 0, 2, 47, 95, 49, 9, 1, 0, 6, 105, 363, 297, 93, 12, 1, 0, 12, 248, 1292, 1660, 753, 158, 16, 1, 0, 11, 563, 4649, 8409, 5591, 1653, 250, 20, 1, 0, 2, 1414, 15976, 41264, 38074, 15590, 3249, 380, 25, 1
Offset: 0

Views

Author

Gus Wiseman, Apr 16 2024

Keywords

Examples

			The set partition {{1,3},{2},{4}} has two distinct block-sums {2,4} so is counted under T(4,2).
Triangle begins:
     1
     0     1
     0     1     1
     0     2     2     1
     0     2     8     4     1
     0     2    19    24     6     1
     0     2    47    95    49     9     1
     0     6   105   363   297    93    12     1
     0    12   248  1292  1660   753   158    16     1
     0    11   563  4649  8409  5591  1653   250    20     1
     0     2  1414 15976 41264 38074 15590  3249   380    25     1
Row n = 4 counts the following set partitions:
  .  {{1,4},{2,3}}  {{1},{2,3,4}}    {{1},{2},{3,4}}  {{1},{2},{3},{4}}
     {{1,2,3,4}}    {{1,2},{3},{4}}  {{1},{2,3},{4}}
                    {{1,2},{3,4}}    {{1},{2,4},{3}}
                    {{1,3},{2},{4}}  {{1,4},{2},{3}}
                    {{1,3},{2,4}}
                    {{1,2,3},{4}}
                    {{1,2,4},{3}}
                    {{1,3,4},{2}}

Crossrefs

Row sums are A000110.
Column k = 1 is A035470.
A version for integer partitions is A116608.
For block lengths instead of sums we have A208437.
A008277 counts set partitions by length.
A275780 counts set partitions with distinct block-sums.
A371737 counts quanimous strict partitions, non-strict A321452.
A371789 counts non-quanimous sets, differences A371790.
Cf. A007837, A038041, A327899, A336137, A336138, A365663, A365661, A365925.

Programs

  • Mathematica
    sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[Range[n]], Length[Union[Total/@#]]==k&]],{n,0,5},{k,0,n}]

A097522 Triangle read by rows giving the 246 multinomials described by A005651(5) related to Young tableau and Kostka numbers.

Original entry on oeis.org

1, 16, 1, 25, 12, 1, 36, 15, 8, 1, 25, 18, 10, 8, 1, 16, 10, 6, 5, 4, 1, 1, 4, 5, 6, 5, 4, 1
Offset: 1

Views

Author

Alford Arnold, Aug 27 2004

Keywords

Comments

The antidiagonal 1 4 5 6 5 4 1 is also listed in A007837 in ascending order by value: 1 1 4 4 5 5 6. The 246 cases are distributed in A036038 as 1 5 10 20 30 60 120.

Examples

			Triangle is
   1;
  16,  1;
  25, 12,  1;
  36, 15,  8,  1;
  25, 18, 10,  8,  1;
  16, 10,  6,  5,  4,  1;
   1,  4,  5,  6,  5,  4,  1;

References

  • R. Stanton, Constructive Combinatorics, 19856, page 83.

Crossrefs

Cf. A000085, A000142, A003869, A005651, A036038, A060240.

A115275 Number of partitions of {1,...,n} into blocks such that no block size is repeated more than 3 times.

Original entry on oeis.org

1, 1, 2, 5, 14, 51, 187, 820, 3670, 18191, 97917, 554500, 3334465, 20871592, 138440031, 972083845, 6985171390, 52194795327, 412903730293, 3313067916192, 28017395030419, 241504438776956, 2189375704925081, 19771679215526507, 187677937412341677
Offset: 0

Views

Author

Christian G. Bower, Jan 18 2006

Keywords

Links

Crossrefs

Cf. A000110, A001935, A007837, A115276, A115277.

Programs

  • Maple
    with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<1, 0, add(multinomial(n, n-i*j, i$j)/j!*
      b(n-i*j, i-1), j=0..min(3, n/i))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Sep 17 2015
  • Mathematica
    multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[multinomial[n, Join[{n-i*j}, Array[i&, j]]]/j!*b[n - i*j, i-1], {j, 0, Min[3, n/i]}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Oct 29 2015, after Alois P. Heinz *)

Formula

E.g.f.: Product {m >= 1} (1+x^m/m!+(x^m/m!)^2+(x^m/m!)^3). [this e.g.f. is incorrect. - Vaclav Kotesovec, Oct 29 2015]

A183229 G.f.: Sum_{n>=0} a(n)*x^n/n!^2 = Product_{n>=1} (1 + x^n/n!^2).

Original entry on oeis.org

1, 1, 1, 10, 17, 126, 3862, 12741, 110609, 1929430, 167593826, 845443941, 11064102326, 178820437901, 7538334414717, 1483432379403435, 10962589471724049, 189591619730952006, 3827839859607324106
Offset: 0

Views

Author

Paul D. Hanna, Jan 04 2011

Keywords

Examples

			G.f.: A(x) = 1 + x + x^2/2!^2 + 10*x^3/3!^2 + 17*x^4/4!^2 +...
A(x) = (1 + x)*(1 + x^2/2!^2)*(1 + x^3/3!^2)*(1 + x^4/4!^2)*...

Crossrefs

Cf. A183230, A007837.

Programs

  • PARI
    {a(n)=n!^2*polcoeff(prod(k=1, n, 1+x^k/k!^2 +x*O(x^n)), n)}
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