A005651 -id:A005651 - cp's OEIS frontend

A327899 Number of set partitions of {1..n} with equal block sizes and equal block sums.

1, 1, 1, 1, 2, 1, 2, 1, 6, 3, 2, 1, 63, 1, 2, 317, 657, 1, 4333, 1, 9609

Offset: 0

Gus Wiseman, Sep 29 2019

			The a(8) = 6 set partitions:
     {{1,2,3,4,5,6,7,8}}
    {{1,2,7,8},{3,4,5,6}}
    {{1,3,6,8},{2,4,5,7}}
    {{1,4,5,8},{2,3,6,7}}
    {{1,4,6,7},{2,3,5,8}}
  {{1,8},{2,7},{3,6},{4,5}}

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Set partitions with equal block-sizes are A038041.
Set partitions with equal block-sums are A035470.
Cf. A000110, A000258, A005651, A007837, A008277, A275780, A300335, A319189, A326512, A326513, A326515, A327908.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[Range[n]],And[SameQ@@Length/@#,SameQ@@Total/@#]&]],{n,0,8}]

A335642 Expansion of e.g.f. Product_{k>0} 1/(1 - sin(x)^k / k!).

Original entry on oeis.org

1, 1, 3, 9, 35, 147, 710, 3780, 21391, 136063, 932190, 6887232, 55902274, 497726270, 4711586833, 47692742905, 528539419087, 6093676850975, 73010887114406, 943925266298096, 12740929019736310, 175037826035276730, 2561985529052306447, 39817440376814520907, 622315443336146270858

Offset: 0

Seiichi Manyama, Oct 03 2020

sign

a(74) is negative. - Vaclav Kotesovec, Oct 04 2020

Cf. A005651, A335626, A335635, A335643, A335644.

max = 24; Range[0, max]! * CoefficientList[Series[Product[1/(1 - Sin[x]^k/k!), {k, 1, max}], {x, 0, max}], x] (* Amiram Eldar, Oct 04 2020 *)

N=40; x='x+O('x^N); Vec(serlaplace(1/prod(k=1, N, 1-sin(x)^k/k!)))

N=40; x='x+O('x^N); Vec(serlaplace(exp(sum(i=1, N, sum(j=1, N\i, sin(x)^(i*j)/(i*j!^i))))))

E.g.f.: exp( Sum_{i>0} Sum_{j>0} sin(x)^(i*j)/(i*(j!)^i) ).

A341505 E.g.f.: Product_{i>=1, j>=1} 1 / (1 - x^(ij) / (ij)!).

Original entry on oeis.org

1, 1, 4, 14, 77, 427, 3076, 23088, 205316, 1936275, 20611750, 233576818, 2909340750, 38527889389, 551372037898, 8364582709282, 135560933977809, 2320127265064403, 42072789623722518, 802547153889643250, 16118882845967168807, 339268639052195731063

Offset: 0

Ilya Gutkovskiy, Feb 13 2021

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..447

Cf. A000005, A005651, A006171, A174661, A318693, A340903, A341506, A341876.

nmax = 21; CoefficientList[Series[Product[1/(1 - x^k/k!)^DivisorSigma[0, k], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, (n - 1)! Sum[Sum[d DivisorSigma[0, d]/(d!)^(k/d), {d, Divisors[k]}] a[n - k]/(n - k)!, {k, 1, n}]]; Table[a[n], {n, 0, 21}]

E.g.f.: Product_{k>=1} 1 / (1 - x^k / k!)^sigma_0(k).

a(n) ~ c * n!, where c = Product_{k>=2} 1/(1 - 1/k!)^sigma_0(k) = 6.6953800201104498115311861820134227776761182601282551253439990653959... - Vaclav Kotesovec, Feb 20 2021

A348704 a(n) = Sum_{x_1+x_2+ ... +x_n=n, 0 <= x_1<= x_2 <= ... <= x_n <= n} ((n-1)n)!/((n-x_1)! (n-x_2)! * ... * (n-x_n)!).

Original entry on oeis.org

1, 1, 3, 170, 1027950, 1079901406584, 448687115051986530720, 89290138377185872821028908288000, 14759276773881730859717740767606565269685350000, 2387650794422480788739162652666454048976136433287918499830000000

Offset: 0

Seiichi Manyama, Oct 30 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..15

Cf. A027306, A092473, A348701.

Cf. A005651, A348703.

def f(n)
  return 1 if n < 2
  (1..n).inject(:*)
end
def A(k, n)
  sum = 0
  m = f((k - 1) * n)
  (0..n).to_a.repeated_combination(k){|i|
    if (0..k - 1).inject(0){|s, j| s + i[j]} == n
      sum += m / (0..k - 1).inject(1){|s, j| s * f(n - i[j])}
    end
  }
  sum
end
def A348704(n)
  (0..n).map{|i| A(i, i)}
end
p A348704(7)

A104779 a(n) is the sum of entries of n-th Kostka matrix for the partitions of n.

Original entry on oeis.org

1, 1, 3, 7, 21, 57, 182, 565, 1931, 6670, 24537, 92337, 364602, 1477148, 6219031, 26875932, 119930947, 548688443, 2580814003, 12425175838, 61302331782, 309055818656, 1592723862598, 8374123173858, 44917765035082, 245452258746785, 1366116578058731, 7736098938006873

Offset: 0

Alford Arnold, Mar 24 2005

a(n) is the number of symmetric nonnegative integer matrices with sum of entries equal to n, no zero rows or columns, and weakly decreasing row and column sums. - Ludovic Schwob, Aug 29 2023

			For n=4, {1,1,1,1,1} + {0,1,1,2,3} + {0,0,1,1,2} + {0,0,0,1,3} + {0,0,0,0,1} = 21.

Ludovic Schwob, Table of n, a(n) for n = 0..39
E. Egge et al., From quasisymmetric expansions to Schur expansions via a modified inverse Kostka matrix
E. Egge et al., From quasisymmetric expansions to Schur expansions via a modified inverse Kostka matrix, European Journal of Combinatorics, Volume 31, Issue 8, December 2010, Pages 2014-2027.
Wouter Meeussen, Schur Polynomials
Wouter Meeussen, Kostka numbers up to partitions of 20
Wouter Meeussen, Mathematica code for 'kostka' function
Ludovic Schwob, On the enumeration of double cosets and self-inverse double cosets, arXiv:2506.04007 [math.CO], 2025. See p. 13.

Cf. A000041, A000085, A005651, A036038, A097522, A104707, A178718.

Row sums of A104778.

Mathematica
```
(* See Meeussen link. *)
```

Row sums of A104778.

a(7) corrected by Alford Arnold, Dec 31 2010

a(8)-a(21) from Amiram Eldar, May 03 2024

A249619 Triangle T(m,n) = number of permutations of a multiset with m elements and signature corresponding to n-th integer partition (A194602).

Original entry on oeis.org

1, 1, 2, 1, 6, 3, 1, 24, 12, 4, 6, 1, 120, 60, 20, 30, 5, 10, 1, 720, 360, 120, 180, 30, 60, 6, 90, 15, 20, 1, 5040, 2520, 840, 1260, 210, 420, 42, 630, 105, 140, 7, 210, 21, 35, 1, 40320, 20160, 6720, 10080, 1680, 3360, 336, 5040, 840, 1120, 56

Offset: 0

Tilman Piesk, Nov 04 2014

This triangle shows the same numbers in each row as A036038 and A078760 (the multinomial coefficients), but in this arrangement the multisets in column n correspond to the n-th integer partition in the infinite order defined by A194602.
Row lengths: A000041 (partition numbers), Row sums: A005651
Columns: 0: A000142 (factorials), 1: A001710, 2: A001715, 3: A133799, 4: A001720, 6: A001725, 10: A001730, 14: A049388
Last in row: end-2: A037955 after 1 term mismatch, end-1: A001405, end: A000012
The rightmost columns form the triangle A173333:
  n       0      1     2     4    6  10  14  21      (A000041(1,2,3...)-1)
m
1         1
2         2      1
3         6      3     1
4        24     12     4     1
5       120     60    20     5    1
6       720    360   120    30    6   1
7      5040   2520   840   210   42   7   1
8     40320  20160  6720  1680  336  56   8   1
A249620 shows the number of partitions of the same multisets. A187783 shows the number of permutations of special multisets.

			Triangle begins:
  n     0    1    2    3   4   5  6   7   8   9 10
m
0       1
1       1
2       2    1
3       6    3    1
4      24   12    4    6   1
5     120   60   20   30   5  10  1
6     720  360  120  180  30  60  6  90  15  20  1

Tilman Piesk, Triangle rows m=0..25, flattened
Tilman Piesk, Illustration of the multisets for m=0..6

Cf. A036038, A078760, A005651, A005651, A249620, A000041, A173333.

A294531 E.g.f.: 1/Product_{k>0} (1-x^(2k-1)/(2k-1)!).

Original entry on oeis.org

1, 1, 2, 7, 28, 141, 866, 6063, 48560, 438721, 4387582, 48272643, 579642328, 7535456657, 105499762356, 1582665820557, 25322712724800, 430488412249937, 7748929638924950, 147229720951176075, 2944597048114831688, 61836721841638907121, 1360407969674984670156

Offset: 0

Seiichi Manyama, Nov 02 2017

nonn

Robert Israel, Table of n, a(n) for n = 0..449

Cf. A005651, A032310.

N:= 40:
S:= series(1/mul(1-x^j/j!,j=1..N,2),x,N+1):
seq(coeff(S,x,n)*n!,n=0..N); # Robert Israel, Nov 23 2017

nmax = 30; CoefficientList[Series[1/Product[(1-x^(2*k-1)/(2*k-1)!), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Nov 02 2017 *)

A306039 Expansion of e.g.f. Product_{k>=1} 1/(1 - log(1 + x)^k/k!).

Original entry on oeis.org

1, 1, 2, 3, 14, 0, 359, -1988, 28706, -312210, 4387572, -62769366, 1006242599, -17203315363, 318393704043, -6296931104285, 133039045075494, -2986262905171914, 71018001954178952, -1783064497977512206, 47133484019671647932, -1308274154275749372040, 38042727898691562357962

Offset: 0

Ilya Gutkovskiy, Jun 17 2018

sign

N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Stirling Transform

Cf. A005651, A140585, A306040.

a:=series(mul(1/(1-log(1+x)^k/k!),k=1..100),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # Paolo P. Lava, Mar 26 2019

nmax = 22; CoefficientList[Series[Product[1/(1 - Log[1 + x]^k/k!), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Exp[Sum[Sum[Log[1 + x]^(j k)/(k (j!)^k), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] Total[Apply[Multinomial, IntegerPartitions[k], {1}]], {k, 0, n}], {n, 0, 22}]

E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} log(1 + x)^(j*k)/(k*(j!)^k)).

a(n) = Sum_{k=0..n} Stirling1(n,k)*A005651(k).

A319174 a(n) = n! * [x^n] Product_{k>=1} 1/(1 - x^k/k!)^n.

Original entry on oeis.org

1, 1, 8, 90, 1448, 29750, 747462, 22182741, 759504720, 29468021238, 1277744462870, 61232148035531, 3213710056592796, 183329936018667035, 11294683874759287030, 747379761629288205795, 52864744954736491460768, 3980505280416276751035270, 317877846102688099315299678

Offset: 0

Ilya Gutkovskiy, Sep 12 2018

nonn

Cf. A005651, A174661, A319175, A319176.

Table[n! SeriesCoefficient[Product[1/(1 - x^k/k!)^n, {k, 1, n}], {x, 0, n}], {n, 0, 18}]
Table[n! SeriesCoefficient[Exp[n Sum[Sum[x^(j k)/(k (j!)^k), {j, 1, n}], {k, 1, n}]], {x, 0, n}], {n, 0, 18}]

a(n) = n! * [x^n] exp(n*Sum_{k>=1} Sum_{j>=1} x^(j*k)/(k*(j!)^k)).

A319192 Irregular triangle where T(n,k) is the coefficient of p(y) in n! * Sum_{i1 < ... < in} (x_i1 * ... * x_in), where p is power-sum symmetric functions and y is the integer partition with Heinz number A215366(n,k).

Original entry on oeis.org

1, -1, 1, 2, -3, 1, -6, 3, 8, -6, 1, 24, -30, -20, 15, 20, -10, 1, -120, 90, 144, 40, -15, -90, -120, 45, 40, -15, 1, 720, -840, -504, -420, 630, 504, 210, 280, -105, -210, -420, 105, 70, -21, 1, -5040, 5760, 3360, 1260, -3360, 2688, -1260, -4032, -3360, -1120

Offset: 1

Gus Wiseman, Sep 13 2018

A generalization of the triangle of Stirling numbers of the first kind, these are the coefficients appearing in the expansion of single-part augmented elementary symmetric functions in terms of power-sum symmetric functions.

			Triangle begins:
   1
  -1   1
   2  -3   1
  -6   3   8  -6   1
  24 -30 -20  15  20 -10   1
The fourth row corresponds to the symmetric function identity: 24 e(4) = -6 p(4) + 3 p(22) + 8 p(31) - 6 p(211) + p(1111).

Other row orderings are A036039 and A102189.

Cf. A000041, A005651, A008277, A008480, A048994, A056239, A124794, A124795, A215366, A318762, A319182, A319191.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
numPermsOfType[ptn_]:=Total[ptn]!/Times@@ptn/Times@@Factorial/@Length/@Split[ptn];
Table[(-1)^(Total[primeMS[m]]-PrimeOmega[m])*numPermsOfType[primeMS[m]],{n,5},{m,Sort[Times@@Prime/@#&/@IntegerPartitions[n]]}]

cp's OEIS Frontend

A327899 Number of set partitions of {1..n} with equal block sizes and equal block sums.

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A335642 Expansion of e.g.f. Product_{k>0} 1/(1 - sin(x)^k / k!).

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A341505 E.g.f.: Product_{i>=1, j>=1} 1 / (1 - x^(i*j) / (i*j)!).

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A348704 a(n) = Sum_{x_1+x_2+ ... +x_n=n, 0 <= x_1<= x_2 <= ... <= x_n <= n} ((n-1)*n)!/((n-x_1)! * (n-x_2)! * ... * (n-x_n)!).

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A104779 a(n) is the sum of entries of n-th Kostka matrix for the partitions of n.

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A249619 Triangle T(m,n) = number of permutations of a multiset with m elements and signature corresponding to n-th integer partition (A194602).

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A294531 E.g.f.: 1/Product_{k>0} (1-x^(2*k-1)/(2*k-1)!).

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A306039 Expansion of e.g.f. Product_{k>=1} 1/(1 - log(1 + x)^k/k!).

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Mathematica

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A319174 a(n) = n! * [x^n] Product_{k>=1} 1/(1 - x^k/k!)^n.

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A341505 E.g.f.: Product_{i>=1, j>=1} 1 / (1 - x^(ij) / (ij)!).

A348704 a(n) = Sum_{x_1+x_2+ ... +x_n=n, 0 <= x_1<= x_2 <= ... <= x_n <= n} ((n-1)n)!/((n-x_1)! (n-x_2)! * ... * (n-x_n)!).

A294531 E.g.f.: 1/Product_{k>0} (1-x^(2k-1)/(2k-1)!).