A209319
Number of functions f:{1,2,...,n}->{1,2,...,n} whose cycle lengths are <= 2.
Original entry on oeis.org
1, 1, 4, 25, 218, 2451, 33832, 554527, 10535100, 227790505, 5525843696, 148673435769, 4394818486456, 141611317636075, 4940870266568160, 185595910032346111, 7468517348971708688, 320562141349559055633, 14619577651630443611200, 706025600924216704982425
Offset: 0
a(3) = 25 because there are 27 functions from {1,2,3} into itself but 2 of these have cycle length of 3: 2,3,1, and 3,1,2.
-
T:= -LambertW(-x):
egf:= exp(T + T^2/2):
a:= n-> n!*coeff(series(egf, x, n+1), x, n):
seq(a(n), n=0..20); # Alois P. Heinz, Jan 19 2013
-
nn=20;t=Sum[n^(n-1)x^n/n!,{n,1,nn}];Range[0,nn]!CoefficientList[Series[Exp[Sum[t^i/i,{i,1,2}]],{x,0,nn}],x]
A246523
Number of endofunctions on [n] whose cycle lengths are divisors of 3.
Original entry on oeis.org
1, 1, 3, 18, 157, 1776, 24687, 407464, 7792857, 169554240, 4137133051, 111912543744, 3324740466357, 107628168419968, 3771341043102375, 142230049514309376, 5744687204783023153, 247424591909961916416, 11320453594446364577907, 548348501001426735001600
Offset: 0
-
with(numtheory):
egf:= k-> exp(add((-LambertW(-x))^d/d, d=divisors(k))):
a:= n-> n!*coeff(series(egf(3), x, n+1), x, n):
seq(a(n), n=0..25);
# second Maple program:
with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1)*
(i-1)!^j, j=0..`if`(irem(3, i)=0, n/i, 0))))
end:
a:= n-> add(b(j, min(3, j))*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(a(n), n=0..25);
A246524
Number of endofunctions on [n] whose cycle lengths are divisors of 4.
Original entry on oeis.org
1, 1, 4, 25, 224, 2601, 37072, 626137, 12227280, 271086625, 6727858496, 184818121929, 5568152828416, 182575550335465, 6473161538599680, 246781048203043561, 10067677495565927168, 437653901985319521153, 20197310874805488471040, 986221173076368356013625
Offset: 0
-
with(numtheory):
egf:= k-> exp(add((-LambertW(-x))^d/d, d=divisors(k))):
a:= n-> n!*coeff(series(egf(4), x, n+1), x, n):
seq(a(n), n=0..25);
# second Maple program:
with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1)*
(i-1)!^j, j=0..`if`(irem(4, i)=0, n/i, 0))))
end:
a:= n-> add(b(j, min(4, j))*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(a(n), n=0..25);
A246525
Number of endofunctions on [n] whose cycle lengths are divisors of 5.
Original entry on oeis.org
1, 1, 3, 16, 125, 1320, 17671, 286336, 5436153, 118144000, 2889312875, 78480441216, 2343333279157, 76274737767424, 2687742759243375, 101931212748928000, 4139544785141163761, 179235455194948829184, 8242391462093927638867, 401202300756829929472000
Offset: 0
-
with(numtheory):
egf:= k-> exp(add((-LambertW(-x))^d/d, d=divisors(k))):
a:= n-> n!*coeff(series(egf(5), x, n+1), x, n):
seq(a(n), n=0..25);
# second Maple program:
with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1)*
(i-1)!^j, j=0..`if`(irem(5, i)=0, n/i, 0))))
end:
a:= n-> add(b(j, min(5, j))*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(a(n), n=0..25);
A246526
Number of endofunctions on [n] whose cycle lengths are divisors of 6.
Original entry on oeis.org
1, 1, 4, 27, 250, 2951, 42552, 726097, 14318908, 320511105, 8029282096, 222590246099, 6765751467576, 223748991426247, 7998566722112800, 307359039816710361, 12634664945078752528, 553260940314226017473, 25711427896197877574208, 1263904006537455579001675
Offset: 0
-
with(numtheory):
egf:= k-> exp(add((-LambertW(-x))^d/d, d=divisors(k))):
a:= n-> n!*coeff(series(egf(6), x, n+1), x, n):
seq(a(n), n=0..25);
# second Maple program:
with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1)*
(i-1)!^j, j=0..`if`(irem(6, i)=0, n/i, 0))))
end:
a:= n-> add(b(j, min(6, j))*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(a(n), n=0..25);
A246527
Number of endofunctions on [n] whose cycle lengths are divisors of 7.
Original entry on oeis.org
1, 1, 3, 16, 125, 1296, 16807, 262864, 4829049, 102073600, 2441582891, 65201946624, 1922453391157, 62009850843136, 2171369477933775, 82007515430081536, 3322113623606686193, 143662773881554108416, 6604529623711334804179, 321608928954695680000000
Offset: 0
-
with(numtheory):
egf:= k-> exp(add((-LambertW(-x))^d/d, d=divisors(k))):
a:= n-> n!*coeff(series(egf(7), x, n+1), x, n):
seq(a(n), n=0..25);
# second Maple program:
with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1)*
(i-1)!^j, j=0..`if`(irem(7, i)=0, n/i, 0))))
end:
a:= n-> add(b(j, min(7, j))*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(a(n), n=0..25);
A246528
Number of endofunctions on [n] whose cycle lengths are divisors of 8.
Original entry on oeis.org
1, 1, 4, 25, 224, 2601, 37072, 626137, 12232320, 271494865, 6750538496, 185923318329, 5619645500416, 184961854976185, 6585429015521280, 252203521861645561, 10338251689510381568, 451650823526438037153, 20949317446607098716160, 1028215744082428119960025
Offset: 0
-
with(numtheory):
egf:= k-> exp(add((-LambertW(-x))^d/d, d=divisors(k))):
a:= n-> n!*coeff(series(egf(8), x, n+1), x, n):
seq(a(n), n=0..25);
# second Maple program:
with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1)*
(i-1)!^j, j=0..`if`(irem(8, i)=0, n/i, 0))))
end:
a:= n-> add(b(j, min(8, j))*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(a(n), n=0..25);
A246529
Number of endofunctions on [n] whose cycle lengths are divisors of 9.
Original entry on oeis.org
1, 1, 3, 18, 157, 1776, 24687, 407464, 7792857, 169594560, 4141165051, 112178655744, 3339749183157, 108422228887168, 3812520677598375, 144372964560581376, 5858088633723823153, 253575577033176047616, 11664031615012086920307, 568166632439929892761600
Offset: 0
-
with(numtheory):
egf:= k-> exp(add((-LambertW(-x))^d/d, d=divisors(k))):
a:= n-> n!*coeff(series(egf(9), x, n+1), x, n):
seq(a(n), n=0..25);
# second Maple program:
with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1)*
(i-1)!^j, j=0..`if`(irem(9, i)=0, n/i, 0))))
end:
a:= n-> add(b(j, min(9, j))*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(a(n), n=0..25);
A246530
Number of endofunctions on [n] whose cycle lengths are divisors of 10.
Original entry on oeis.org
1, 1, 4, 25, 218, 2475, 34696, 579223, 11220540, 247395097, 6117023600, 167639670441, 5044046990776, 165322086357715, 5863394794421088, 223751099288794375, 9141963589243198736, 398198217292835137137, 18420080017512816009280, 901874615547758970425977
Offset: 0
-
with(numtheory):
egf:= k-> exp(add((-LambertW(-x))^d/d, d=divisors(k))):
a:= n-> n!*coeff(series(egf(10), x, n+1), x, n):
seq(a(n), n=0..25);
# second Maple program:
with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1)*
(i-1)!^j, j=0..`if`(irem(10, i)=0, n/i, 0))))
end:
a:= n-> add(b(j, min(10, j))*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(a(n), n=0..25);
A246531
Number of endofunctions on [n] whose cycle lengths are divisors of n.
Original entry on oeis.org
1, 1, 4, 18, 224, 1320, 42552, 262864, 12232320, 169594560, 6117023600, 61920993024, 8022787347456, 56694391376896, 5193025319432160, 174746314698336000, 10338252997184749568, 121439552019384139776, 26096843176349347142208, 262144006402373705728000
Offset: 0
-
with(numtheory):
egf:= k-> exp(add((-LambertW(-x))^d/d, d=divisors(k))):
a:= n-> n!*coeff(series(egf(n), x, n+1), x, n):
seq(a(n), n=0..20);
# second Maple program:
with(combinat):
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1, k)*
(i-1)!^j, j=0..`if`(irem(k, i)=0, n/i, 0))))
end:
a:= n-> add(b(j$2, n)*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(a(n), n=0..20);
-
multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0,
Sum[multinomial[n, Join[{n - i*j},
Table[i, {j}]]]/j!*b[n - i*j, i - 1, k]*(i - 1)!^j,
{j, 0, If[Mod[k, i] == 0, n/i, 0]}]]];
a[n_] := If[n==0, 1, Sum[b[j, j, n]*n^(n-j)*Binomial[n-1, j-1], {j, 0, n}]];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, May 01 2022, after Alois P. Heinz *)
Showing 1-10 of 10 results.