A364344
Number of endofunctions on [n] such that the number of elements that are mapped to i is a multiple or a divisor of i.
Original entry on oeis.org
1, 1, 4, 20, 177, 1462, 21919, 254802, 4816788, 82401465, 1929926410, 35256890748, 1152938630784, 24977973856643, 823036511854847, 24332827884557037, 954801492779273665, 27023410818058291822, 1309814517293654535339, 41375530521928893861920
Offset: 0
a(0) = 1: ().
a(1) = 1: (1).
a(2) = 2: (11), (12), (21), (22).
a(3) = 20 (111), (112), (113), (121), (122), (123), (131), (132), (211), (212), (213), (221), (223), (231), (232), (311), (312), (321), (322), (333).
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b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(b(n-i*j, i-1)*binomial(n, i*j), j=0..n/i)+add(
`if`(d>n or d=i, 0, b(n-d, i-1)*binomial(n, d)),
d=numtheory[divisors](i))))
end:
a:= n-> b(n$2):
seq(a(n), n=0..19);
-
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[b[n - i*j, i - 1]* Binomial[n, i*j], {j, 0, n/i}]+Sum[If[d>n || d == i, 0, b[n - d, i - 1]* Binomial[n, d]], {d, Divisors[i]}]]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Oct 27 2023, after Alois P. Heinz *)
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);
A351991
Expansion of e.g.f. Product_{k>=1} 1/(1 - x^prime(k) / prime(k)!).
Original entry on oeis.org
1, 0, 1, 1, 6, 11, 110, 232, 3136, 10032, 141492, 561001, 9708864, 43864471, 886873780, 4775054571, 106484438048, 651701988633, 16430255716392, 111498817293652, 3124376125290804, 23598473924038152, 721819990948712614, 5975067276022072669, 199594173117826648968
Offset: 0
A371550
Expansion of e.g.f. Product_{k>=1} (1 + mu(k)^2*x^k/k!).
Original entry on oeis.org
1, 1, 1, 4, 4, 11, 67, 29, 260, 876, 3841, 34134, 69774, 152231, 774243, 4182754, 30376720, 409813561, 1056300594, 3175397668, 3655126844, 91668397027, 499871922705, 5219638765816, 120716816247428, 17518596045460, 193032439198301, 3860666483055372, 22675538336311998
Offset: 0
-
nmax = 28; CoefficientList[Series[Product[(1 + MoebiusMu[k]^2 x^k/k!), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
Showing 1-5 of 5 results.