A273442 Number of endofunctions on [2n] with exactly n cycles.

1, 3, 95, 5595, 484729, 55545735, 7923937307, 1353285904971, 269240651261153, 61157779792168059, 15617503320899550135, 4429016799173481942427, 1381112305978592892946825, 469689278931628969590283855, 173002815169302537782725771395

Offset

0,2

Links

Alois P. Heinz, Table of n, a(n) for n = 0..288

Formula

a(n) = (2*n)!/n! * [x^(2*n)] (-log(1+LambertW(-x)))^n.

a(n) = A060281(2n,n).

a(n) ~ c * d^n * n^(n-1/2), where d = 2^(4-r) * exp(1-r) * (2-r)^(r-2) * log(s) / (1-1/s)^r = 10.40858458700790823344027277763248832..., where r = 1.2672171362228848078038115564503589940694831794020694762759870935... is the root of the equation r*log(s) * (-1 + (r-s)* log((2*(s-1))/(s*(2-r)))) = 1 - s, where s = -r*LambertW(-1, -exp(-1/r)/r) = 1.5782614856055967129193228312616913... and c = 0.336740238865974324583136447665761... - Vaclav Kotesovec, Nov 01 2016, extended Aug 28 2017

Mathematica

Table[(2*n)!/n! * SeriesCoefficient[(-Log[1+LambertW[-x]])^n, {x, 0, 2*n}], {n, 0, 20}] (* Vaclav Kotesovec, Nov 01 2016 *)
Flatten[{1, Table[Sum[Binomial[2*n-1, k] * (2*n)^(2*n-1-k) * Abs[StirlingS1[k+1, n]], {k, 0, 2*n-1}], {n, 1, 20}]}] (* Vaclav Kotesovec, Nov 01 2016 *)

Crossrefs

Cf. A060281.

Sequence in context: A241752 A093009 A373551 * A322460 A368012 A249787

Adjacent sequences: A273439 A273440 A273441 * A273443 A273444 A273445

Keyword

nonn

Author

Alois P. Heinz, May 22 2016

Status

approved