A185025 - OEIS

A185025

Triangular array read by rows: T(n,k) is the number of functions f:{1,2,...,n} -> {1,2,...,n} that have exactly k 2-cycles for n >= 0 and 0 <= k <= floor(n/2).

1, 1, 3, 1, 18, 9, 163, 90, 3, 1950, 1100, 75, 28821, 16245, 1575, 15, 505876, 283122, 33810, 735, 10270569, 5699932, 780150, 26460, 105, 236644092, 130267440, 19615932, 884520, 8505, 6098971555, 3332614725, 538325550, 29619450, 467775, 945

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OFFSET

0,3

COMMENTS

It appears that as n gets large, row n conforms to a Poisson distribution with mean = 1/2. In other words, as n gets large, T(n,k) approaches n^n/(2^k*k!*e^(1/2)).

LINKS

Table of n, a(n) for n=0..35.

FORMULA

E.g.f.: exp((T(x)^2/2)*(y-1))/(1 - T(x)) where T(x) is the e.g.f. for A000169.

Sum_{k=1..floor(n/2)} k * T(n,k) = A081131(n).

EXAMPLE

Triangle begins:

3, 1;

18, 9;

163, 90, 3;

1950, 1100, 75;

28821, 16245, 1575, 15;