A132958
a(n) = n!*Sum_{d|n} (-1)^(d+1)/d!.
Original entry on oeis.org
1, 1, 7, 11, 121, 479, 5041, 18479, 423361, 1844639, 39916801, 298710719, 6227020801, 43606442879, 1536517382401, 9589093113599, 355687428096001, 4259374594675199, 121645100408832001, 1135353600039859199
Offset: 1
-
f[n_] := Block[{d = Divisors@n}, Plus @@ (n!*(-1)^(d + 1)/d!)]; Array[f, 19] (* or *) (* Robert G. Wilson v, Sep 13 2007 *)
Rest[ Range[0, 20]! CoefficientList[ Series[ Sum[(-x)^k/(k!*(x^k - 1)), {k, 25}], {x, 0, 20}], x]] (* or *) (* Robert G. Wilson v, Sep 13 2007 *)
Rest[ Range[0, 20]! CoefficientList[ Series[ Sum[1 - Exp[ -x^k], {k, 25}], {x, 0, 20}], x]] (* Robert G. Wilson v, Sep 13 2007 *)
-
a(n) = n!*sumdiv(n, d, (-1)^(d+1)/d!); \\ Michel Marcus, Sep 29 2017
A132960
a(n) = n!*Sum_{d|n} (-1)^(d+1)/(d!*(n/d)^d).
Original entry on oeis.org
1, 0, 3, 2, 25, 94, 721, 3674, 42561, 291248, 3628801, 34254604, 479001601, 5337581534, 88966701825, 1140807642974, 20922789888001, 321094542593824, 6402373705728001, 109338195253235948, 2457732174030848001
Offset: 1
-
Rest[ Range[0, 22]! CoefficientList[ Series[ - Sum[ Exp[ -x^k/k], {k, 25}], {x, 0, 22}], x]] (* Robert G. Wilson v, Sep 13 2007 *)
-
a(n) = n!*sumdiv(n, d, (-1)^(d+1)/(d!*(n/d)^d)); \\ Michel Marcus, Sep 29 2017
A132962
a(n) = n!*Sum_{d|n} (-1)^(d+1)/(d!*(n/d)!^d).
Original entry on oeis.org
1, 0, 2, -3, 2, 5, 2, -140, 282, 819, 2, -20482, 2, 133419, 1527528, -4661085, 2, -153296429, 2, 1402482796, 36278688162, 13748957859, 2, -14081800718427, 5194672859378, 7905848380325, 2977584150505252, 12956452725792600, 2, -1314647260913859151
Offset: 1
-
Rest[ Range[0, 30]! CoefficientList[ Series[ Sum[1 - Exp[ -x^k/k! ], {k, 30}], {x, 0, 30}], x]] (* Robert G. Wilson v, Sep 13 2007 *)
-
a(n) = n!*sumdiv(n, d, (-1)^(d+1)/(d!*(n/d)!^d)); \\ Michel Marcus, Sep 29 2017
A132959
Total number of all distinct list sizes in all partitions of [n] into lists, cf. A000262.
Original entry on oeis.org
1, 3, 19, 109, 881, 7621, 77785, 854225, 10750465, 143737381, 2121714761, 33426065905, 568250246305, 10242445089605, 197388381934801, 4003553262384961, 86010508861504385, 1939950117886565125
Offset: 1
-
Rest[ Range[0, 20]! CoefficientList[ Series[ Exp[x/(1 - x)] Sum[(-x)^k/(k!*(x^k - 1)), {k, 25}], {x, 0, 20}], x]] (* Robert G. Wilson v, Sep 13 2007 *)
A132961
Total number of all distinct cycle sizes in all permutations of [n].
Original entry on oeis.org
1, 2, 9, 38, 215, 1384, 10409, 86946, 825075, 8541998, 97590779, 1205343952, 16148472977, 231416203212, 3560209750005, 58104163643054, 1008693571819919, 18477578835352366, 357476371577422955, 7258865626801695048, 154893910336866444009, 3454112338490001478772
Offset: 1
-
with(combinat):
b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,
add(multinomial(n, n-i*j, i$j)/j!*(i-1)!^j*(p-> p+
[0, p[1]*`if`(j>0, 1, 0)])(b(n-i*j, i-1)), j=0..n/i)))
end:
a:= n-> b(n$2)[2]:
seq(a(n), n=1..30); # Alois P. Heinz, Oct 21 2015
-
Rest[ Range[0, 22]! CoefficientList[ Series[1/(1 - x) Sum[1 - Exp[ -x^k/k], {k, 25}], {x, 0, 22}], x]] (* Robert G. Wilson v, Sep 13 2007 *)
A350175
Sum of the distinct block sizes over all partitions of [n].
Original entry on oeis.org
0, 1, 3, 13, 45, 196, 888, 4383, 22879, 129163, 768913, 4849912, 32202712, 224672241, 1640679589, 12517008985, 99484656169, 822410210044, 7055883373604, 62730142658947, 576984726864147, 5482889832932123, 53757450049841167, 543169144098559606, 5649499728403949184
Offset: 0
a(3) = 13 = 1*3 + 3*(1+2) + 1: 123, 1|23, 13|2, 12|3, 1|2|3.
-
b:= proc(n, i, c) option remember; `if`(n=0, c,
`if`(i<1, 0, add(b(n-j*i, i-1, c+i*signum(j))*
combinat[multinomial](n, n-i*j, i$j)/j!, j=0..n/i)))
end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..30);
-
multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, c_] := b[n, i, c] = If[n == 0, c,
If[i < 1, 0, Sum[b[n - j*i, i - 1, c + i*Sign[j]]*
multinomial[n, Join[{n - i*j}, Table[i, {j}]]]/j!, {j, 0, n/i}]]];
a[n_] := b[n, n, 0];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jan 11 2022, after Alois P. Heinz *)
Showing 1-6 of 6 results.