A384504 a(n) = Stirling1(n^2, n).

1, 1, 11, 118124, 5056995703824, 2677503356427960382362624, 43103055200236892507668550744976954163200, 44206966751754314698168885550132827351582613259130314424320000

Offset

0,3

Comments

a(n) is the number of permutations of n^2 objects with n cycles.

Links

Table of n, a(n) for n=0..7.

Formula

a(n)^(1/n^2) ~ exp(-1)*n^2.

a(n) ~ n^((n-1)*(3*n+1)) * w^(n^2) / (sqrt(2*Pi*(w-1)) * exp(n*(n-1)) * (n*w-1)^(n*(n-1))), where w = -LambertW(-1, -exp(-1/n)/n).

Mathematica

Table[StirlingS1[n^2, n], {n, 0, 10}]

Crossrefs

Cf. A008275, A048994, A218141.

Sequence in context: A079558 A111478 A300196 * A055311 A116622 A390248

Adjacent sequences: A384501 A384502 A384503 * A384505 A384506 A384507

Keyword

nonn

Author

Vaclav Kotesovec, May 31 2025

Status

approved