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
A293211
Triangle T(n,k) is the number of permutations on n elements with at least one k-cycle for 1 <= k <= n.
Original entry on oeis.org
1, 1, 1, 4, 3, 2, 15, 9, 8, 6, 76, 45, 40, 30, 24, 455, 285, 200, 180, 144, 120, 3186, 1995, 1400, 1260, 1008, 840, 720, 25487, 15855, 11200, 8820, 8064, 6720, 5760, 5040, 229384, 142695, 103040, 79380, 72576, 60480, 51840, 45360, 40320, 2293839, 1427895, 1030400, 793800, 653184, 604800, 518400, 453600, 403200, 362880
Offset: 1
T(n,k) (the first 8 rows):
: 1;
: 1, 1;
: 4, 3, 2;
: 15, 9, 8, 6;
: 76, 45, 40, 30, 24;
: 455, 285, 200, 180, 144, 120;
: 3186, 1995, 1400, 1260, 1008, 840, 720;
: 25487, 15855, 11200, 8820, 8064, 6720, 5760, 5040;
...
T(4,3)=8 since there are exactly 8 permutations on {1,2,3,4} with at least one 3-cycle: (1)(234), (1)(243), (2)(134), (2)(143), (3)(124), (3)(142), (4)(123), and (4)(132).
-
T:=(n,k)->n!*sum((-1)^(j+1)*(1/k)^j/j!,j=1..floor(n/k)); seq(seq(T(n,k),k=1..n),n=1..10);
-
Table[n!*Sum[(-1)^(j + 1)*(1/k)^j/j!, {j, Floor[n/k]}], {n, 10}, {k, n}] // Flatten (* Michael De Vlieger, Oct 02 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 *)
A132963
Total number of distinct block sizes in all partitions of [n].
Original entry on oeis.org
1, 2, 8, 25, 102, 439, 2067, 10406, 56754, 328257, 2015818, 13067366, 89192170, 638321285, 4779442602, 37332643831, 303635437532, 2565592977205, 22483754207839, 204013083946460, 1913880812797792, 18536832515581167, 185130415180288134, 1904280138346826637
Offset: 1
-
b:= proc(n, i, c) option remember; `if`(n=0, c,
`if`(i<1, 0, add(b(n-j*i, i-1, c+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=1..30); # Alois P. Heinz, Jan 06 2022
-
Rest[ Range[0, 23]! CoefficientList[ Series[ Exp[ Exp[x] - 1] Sum[1 - Exp[ -x^k/k! ], {k, 30}], {x, 0, 23}], x]] (* Robert G. Wilson v, Sep 13 2007 *)
Showing 1-6 of 6 results.
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