A364327
Number of endofunctions on [n] such that the number of elements that are mapped to i is either 0 or a divisor of i.
Original entry on oeis.org
1, 1, 3, 13, 115, 851, 13431, 144516, 2782571, 47046307, 1107742273, 19263747713, 657152726011, 13657313316986, 451605697223110, 13377063396461138, 531234399267707419, 14563460779785318719, 721703507708044677945, 22141894282020163910406, 1123287408943765640907425
Offset: 0
a(0) = 1: ().
a(1) = 1: (1).
a(2) = 3: (22), (21), (12).
a(3) = 13: (333), (322), (232), (223), (321), (231), (213), (312), (132), (123), (221), (212), (122).
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b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+add(
`if`(d>n, 0, b(n-d, i-1)*binomial(n, d)), d=numtheory[divisors](i))))
end:
a:= n-> b(n$2):
seq(a(n), n=0..23);
A364328
Number of endofunctions on [n] such that the number of elements that are mapped to i is either 0 or a prime divisor of i.
Original entry on oeis.org
1, 0, 1, 1, 6, 21, 110, 904, 4312, 74400, 731412, 5600761, 128196024, 792051157, 18696610816, 264267572121, 7136433698464, 57948743342529, 2228312959187256, 22463157401776612, 681974906329502904, 15395459281239915282, 463374873030990445252, 6091833036158810701465
Offset: 0
a(0) = 1: ().
a(2) = 1: (22).
a(3) = 1: (333).
a(4) = 6: (4422), (4242), (4224), (2442), (2424), (2244).
a(5) = 21: (55555), (44333), (43433), (43343), (43334), (34433), (34343), (34334), (33443), (33434), (33344), (33322), (33232), (33223), (32332), (32323), (32233), (23332), (23323), (23233), (22333).
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b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+add(
`if`(d>n, 0, b(n-d, i-1)*binomial(n, d)), d=numtheory[factorset](i))))
end:
a:= n-> b(n$2):
seq(a(n), n=0..23);
A330074
Expansion of e.g.f. Product_{k>=1} (1 - log(1 - x^k)).
Original entry on oeis.org
1, 1, 3, 14, 78, 544, 4560, 42468, 451584, 5382144, 69737760, 985265280, 15204119040, 249602065920, 4398839827200, 82834744849920, 1646970433920000, 34626184595251200, 769149445849989120, 17896198498368583680, 437123791096022016000, 11171177571932111462400
Offset: 0
-
nmax = 21; CoefficientList[Series[Product[1 - Log[1 - x^k], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[Product[(1 + Sum[x^(i j)/i, {i, 1, nmax}]), {j, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
A330075
Expansion of e.g.f. Product_{k>=1} (1 - log(1 - x^k) / k).
Original entry on oeis.org
1, 1, 2, 7, 32, 168, 1184, 8622, 77216, 747576, 8185392, 93054960, 1264465872, 16974221184, 254355732864, 4069961945280, 70258008510720, 1228263760984320, 24025502406873600, 470522155226595840, 10095034628228958720, 222277023267825254400, 5144511652272759029760
Offset: 0
-
nmax = 22; CoefficientList[Series[Product[1 - Log[1 - x^k]/k, {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Product[(1 + Sum[x^(i j)/(i j), {i, 1, nmax}]), {j, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!