A343542 -id:A343542 - cp's OEIS frontend

A341283 Number of ways to partition n labeled elements into sets of different sizes of at least 3.

1, 0, 0, 1, 1, 1, 1, 36, 57, 211, 331, 958, 29228, 64065, 294659, 1232479, 3549717, 11296603, 557617987, 1512758550, 8514685860, 41183585167, 251022906729, 838303110637, 4183056225010, 263978773601641, 887708421995331, 5813843897797861, 32212405278588967, 216518890998518716

Offset: 0

Ilya Gutkovskiy, Apr 28 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..699

Cf. A006505, A007837, A025148, A032311, A102233, A343319, A343542.

b:= proc(n, i) option remember; `if`(n=0, 1,
     `if`(i>n, 0, b(n, i+1)+b(n-i, i+1)*binomial(n, i)))
    end:
a:= n-> b(n, 3):
seq(a(n), n=0..30);  # Alois P. Heinz, Apr 28 2021

nmax = 29; CoefficientList[Series[Product[(1 + x^k/k!), {k, 3, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -(n - 1)! Sum[DivisorSum[k, # (-#!)^(-k/#) &, # > 2 &] a[n - k]/(n - k)!, {k, 1, n}]; Table[a[n], {n, 0, 29}]

E.g.f.: Product_{k>=3} (1 + x^k/k!).

A343319 Number of ways to partition n labeled elements into sets of different sizes of at least 4.

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 1, 1, 1, 127, 211, 793, 1288, 3719, 6007, 646439, 1467077, 7211843, 30123763, 91160937, 293184840, 1118980377, 110635063749, 319072758997, 1918239941962, 9518126978941, 58119248603131, 202992067559011, 1031021295578251, 4151156602678042, 650225250329137612

Offset: 0

Ilya Gutkovskiy, Apr 28 2021

nonn

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

Cf. A007837, A025149, A032311, A057837, A232475, A341283, A343542.

b:= proc(n, i) option remember; `if`(n=0, 1,
     `if`(i>n, 0, b(n, i+1)+b(n-i, i+1)*binomial(n, i)))
    end:
a:= n-> b(n, 4):
seq(a(n), n=0..30);  # Alois P. Heinz, Apr 28 2021

nmax = 30; CoefficientList[Series[Product[(1 + x^k/k!), {k, 4, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -(n - 1)! Sum[DivisorSum[k, # (-#!)^(-k/#) &, # > 3 &] a[n - k]/(n - k)!, {k, 1, n}]; Table[a[n], {n, 0, 30}]

E.g.f.: Product_{k>=4} (1 + x^k/k!).

cp's OEIS Frontend

A341283 Number of ways to partition n labeled elements into sets of different sizes of at least 3.

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