A057814 -id:A057814 - cp's OEIS frontend

A343542 Number of ways to partition n labeled elements into sets of different sizes of at least 5.

1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 463, 793, 3004, 5006, 14444, 23817, 62323, 14805403, 35175993, 177791475, 745977222, 2333540804, 7589340982, 29027728612, 81515120641, 23232813583331, 69799133324911, 436678552247551, 2215090972333651, 13529994077951557, 48863594588239153

Offset: 0

Ilya Gutkovskiy, Apr 28 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..700

Cf. A007837, A025150, A032311, A057814, A245790, A341283, A343319.

b:= proc(n, i) option remember; `if`(n=0, 1,
     `if`(i>n, 0, b(n, i+1)+binomial(n, i)*b(n-i, i+1)))
    end:
a:= n-> b(n, 5):
seq(a(n), n=0..31);  # Alois P. Heinz, Apr 28 2021

nmax = 31; CoefficientList[Series[Product[(1 + x^k/k!), {k, 5, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -(n - 1)! Sum[DivisorSum[k, # (-#!)^(-k/#) &, # > 4 &] a[n - k]/(n - k)!, {k, 1, n}]; Table[a[n], {n, 0, 31}]

E.g.f.: Product_{k>=5} (1 + x^k/k!).

A182931 Generalized Bell numbers; square array read by ascending antidiagonals, A(n, k) for n >= 0 and k >= 1.

Original entry on oeis.org

1, 1, 1, 2, 0, 1, 5, 1, 0, 1, 15, 1, 0, 0, 1, 52, 4, 1, 0, 0, 1, 203, 11, 1, 0, 0, 0, 1, 877, 41, 1, 1, 0, 0, 0, 1, 4140, 162, 11, 1, 0, 0, 0, 0, 1, 21147, 715, 36, 1, 1, 0, 0, 0, 0, 1, 115975, 3425, 92, 1, 1, 0, 0, 0, 0, 0, 1, 678570, 17722, 491, 36, 1, 1, 0, 0, 0, 0, 0, 1

Offset: 1

Peter Luschny, Apr 05 2011

			Array starts:
[k=      1       2       3       4       5]
[n=0]    1,      1,      1,      1,      1,
[n=1]    1,      0,      0,      0,      0,
[n=2]    2,      1,      0,      0,      0,
[n=3]    5,      1,      1,      0,      0,
[n=4]   15,      4,      1,      1,      0,
[n=5]   52,     11,      1,      1,      1,
[n=6]  203,     41,     11,      1,      1,
[n=7]  877,    162,     36,      1,      1,
[n=8] 4140,    715,     92,     36,      1,
   A000110,A000296,A006505,A057837,A057814, ...

E. A. Enneking and J. C. Ahuja, Generalized Bell numbers, Fib. Quart., 14 (1976), 67-73.
Peter Luschny, Set partitions

Cf. A000110, A000296, A006505, A057837, A057814, A097147.

Row sums are A097147 for n >= 1.

egf := k -> exp(exp(x)*(1-GAMMA(k,x)/GAMMA(k)));
T := (n,k) -> n!*coeff(series(egf(k),x,n+1),x,n):
seq(print(seq(T(n,k),k=1..8)),n=0..8);

egf[k_] := Exp[Exp[x] (1 - Gamma[k, x]/Gamma[k])];
T[n_, k_] := n! SeriesCoefficient[egf[k], {x, 0, n}];
Table[T[n-k+1, k], {n, 0, 11}, {k, 1, n+1}] // Flatten (* Jean-François Alcover, Aug 13 2019 *)

E.g.f.: exp(exp(x)*(1-Gamma(k,x)/Gamma(k))); Gamma(k,x) the incomplete Gamma function.

A282988 Triangle of partitions of an n-set into boxes of size >= m.

Original entry on oeis.org

1, 2, 1, 5, 1, 1, 15, 4, 1, 1, 52, 11, 1, 1, 1, 203, 41, 11, 1, 1, 1, 877, 162, 36, 1, 1, 1, 1, 4140, 715, 92, 36, 1, 1, 1, 1, 21147, 3425, 491, 127, 1, 1, 1, 1, 1, 115975, 17722, 2557, 337, 127, 1, 1, 1, 1, 1, 678570, 98253, 11353, 793, 463, 1, 1, 1, 1, 1, 1

Offset: 1

Vladimir Kruchinin, Feb 26 2017

			Triangle T(n,m) begins:
    1;
    2,   1;
    5,   1,   1;
   15,   4,   1,   1;
   52,  11,   1,   1,   1;
  203,  41,  11,   1,   1,   1;
  877, 162,  36,   1,   1,   1,   1;
  ...

Alois P. Heinz, Rows n = 1..141, flattened

Cf. A000110, A000296, A006505, A057837, A057814, A182931, A260878.

T:= proc(n, k) option remember; `if`(n=0, 1, add(
      T(n-j, k)*binomial(n-1, j-1), j=k..n))
    end:
seq(seq(T(n, k), k=1..n), n=1..14);  # Alois P. Heinz, Sep 28 2017

T[n_, m_] := T[n, m] = Which[Or[n == m, n == 0], 1, m == 0, 0, True, Sum[Binomial[n - 1, i + m - 1] T[n - i - m, m], {i, 0, n - m}]]; Table[T[n, m], {n, 11}, {m, n}] // Flatten (* Michael De Vlieger, Feb 26 2017 *)

T(n,m):=if n=m or n=0 then 1 else if m=0 then 0 else sum(binomial(n-1, i+m-1)*T(n-i-m,m), i, 0, n-m);

T(n,m) = Sum_{i=0..n-m} C(n-1, i+m-1)*T(n-i-m, m).

E.g.f. m column of T(n,m) is exp(exp(x)-Sum_{k=0..m} 1/k!x^k).

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A343542 Number of ways to partition n labeled elements into sets of different sizes of at least 5.

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A182931 Generalized Bell numbers; square array read by ascending antidiagonals, A(n, k) for n >= 0 and k >= 1.

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A282988 Triangle of partitions of an n-set into boxes of size >= m.

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