A110468
a(n) = (2*n + 1)!/(n + 1).
Original entry on oeis.org
1, 3, 40, 1260, 72576, 6652800, 889574400, 163459296000, 39520825344000, 12164510040883200, 4644631106519040000, 2154334728240414720000, 1193170003333152768000000, 777776389315596582912000000, 589450799582646796969574400000, 513927415886120176107847680000000
Offset: 0
A346066
Sum of GCD of cycle lengths over all permutations of [n].
Original entry on oeis.org
0, 1, 3, 10, 45, 216, 1505, 9360, 84105, 730240, 7715169, 76204800, 1090114025, 11975040000, 185501455425, 2791872219136, 45361870178625, 690452066304000, 14415096609538625, 236887827111936000, 5448878874163974249, 108418310412206080000, 2381309423564793710625
Offset: 0
a(3) = 10 = 3+3+1+1+1+1: (123), (132), (1)(23), (13)(2), (12)(3), (1)(2)(3).
-
b:= proc(n, g) option remember; `if`(n=0, g, add((j-1)!
*b(n-j, igcd(g, j))*binomial(n-1, j-1), j=1..n))
end:
a:= n-> b(n, 0):
seq(a(n), n=0..24);
-
b[n_, g_] := b[n, g] = If[n == 0, g, Sum[(j - 1)!*
b[n - j, GCD[g, j]]*Binomial[n - 1, j - 1], {j, 1, n}]];
a[n_] := b[n, 0];
Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Mar 06 2022, after Alois P. Heinz *)
A346086
Number of permutations of [2n] such that the GCD of the cycle lengths equals 2.
Original entry on oeis.org
0, 1, 3, 105, 4725, 530145, 45270225, 12034447425, 2116670180625, 737902583042625, 219604524727012425, 137952599116097390625, 49583382753435146240625, 46991310794950147391390625, 25508895927267586991297765625, 24661803286201363305440202410625
Offset: 0
a(1) = 1: (12).
a(2) = 3: (12)(34), (13)(24), (14)(23).
Bisection of column k=2 of
A346085 (even part).
-
b:= proc(n, g) option remember; `if`(n=0, `if`(g=2, 1, 0), `if`(g=1, 0,
add(b(n-j, igcd(g, j))*binomial(n-1, j-1)*(j-1)!, j=2..n)))
end:
a:= n-> b(2*n, 0):
seq(a(n), n=0..19);
-
b[n_, g_] := b[n, g] = If[n == 0, If[g == 2, 1, 0], If[g == 1, 0,
Sum[b[n - j, GCD[g, j]]*Binomial[n - 1, j - 1]*(j - 1)!, {j, 2, n}]]];
a[n_] := b[2*n, 0];
Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Mar 06 2022, after Alois P. Heinz *)
A359951
Number of permutations of [n] such that the GCD of the cycle lengths is a prime.
Original entry on oeis.org
0, 0, 1, 2, 3, 24, 145, 720, 4725, 22400, 602721, 3628800, 67692625, 479001600, 12924021825, 103953833984, 2116670180625, 20922789888000, 959231402754625, 6402373705728000, 257071215652932681, 3242340687872000000, 142597230222616430625, 1124000727777607680000
Offset: 0
a(2) = 1: (12).
a(3) = 2: (123), (132).
a(4) = 3: (12)(34), (13)(24), (14)(23).
a(5) = 24: (12345), (12354), (12435), (12453), (12534), (12543), (13245), (13254), (13425), (13452), (13524), (13542), (14235), (14253), (14325), (14352), (14523), (14532), (15234), (15243), (15324), (15342), (15423), (15432).
-
b:= proc(n, g) option remember; `if`(n=0, `if`(isprime(g), 1, 0),
add(b(n-j, igcd(j, g))*(n-1)!/(n-j)!, j=1..n))
end:
a:= n-> b(n, 0):
seq(a(n), n=0..23);
-
b[n_, g_] := b[n, g] = If[n == 0, If[PrimeQ[g], 1, 0], Sum[b[n - j, GCD[j, g]]*(n - 1)!/(n - j)!, {j, 1, n}]];
a[n_] := b[n, 0];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Dec 13 2023, after Alois P. Heinz *)
Showing 1-4 of 4 results.
Comments