A341283
Number of ways to partition n labeled elements into sets of different sizes of at least 3.
Original entry on oeis.org
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
-
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}]
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
-
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}]
A343542
Number of ways to partition n labeled elements into sets of different sizes of at least 5.
Original entry on oeis.org
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
-
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}]
A371485
Expansion of e.g.f. Product_{k>=2} 1 / (1 - x^k/k!).
Original entry on oeis.org
1, 0, 1, 1, 7, 11, 126, 267, 3655, 11503, 169258, 654413, 11623910, 52505961, 1066163983, 5721040860, 128827399823, 783999460951, 19881737827434, 134931439956945, 3784646604928402, 28564669112399283, 875600527787948801, 7239530824941958612, 242133074649322025674
Offset: 0
-
nmax = 24; CoefficientList[Series[Product[1/(1 - x^k/k!), {k, 2, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
A371493
Expansion of e.g.f. Product_{k>=2} (1 + x^k/k).
Original entry on oeis.org
1, 0, 1, 2, 6, 44, 210, 1644, 11088, 119664, 1034640, 12372480, 139629600, 1877722560, 25389131040, 395162832960, 6041860070400, 105872058754560, 1864694944465920, 35822359116149760, 705399920144640000, 15048474234019430400, 324762706938629836800, 7566557300438795366400
Offset: 0
-
nmax = 23; CoefficientList[Series[Product[(1 + x^k/k), {k, 2, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
Showing 1-5 of 5 results.