A395672 - OEIS

%I #6 May 07 2026 20:00:57

%S 0,0,0,17,0,0,2890,84490,1493688,32186168,810545040,23163353400,

%T 720663872500,24332866501372,887558454336372,34810380537375170,

%U 1460425852562219760,65255195022227232480,3093688414572173306128,155103564618621456778168,8198969255720346583303380

%N Number of endofunctions f:{1,2,...,n}-> {1,2,...,n} whose smallest weakly connected component has exactly 3 nodes.

%C a(n) is always divisible by A001865(3) = 17.

%F E.g.f.: exp(-x-3*x^2/2)*(1-exp(-17*x^3/6)) / (1+LambertW(-x)).

%F Limit_{n->oo} a(n) / n^n = exp(-1/e - 3/(2*e^2)) * (1 - exp(-17/(6*e^3))).

%F a(n) = 17*Sum_{k=0..n} binomial(n,k)*(-1)^(k+1)*(n-k)^(n-k)*Sum_{j=1..floor(k/3)} binomial(k,3*j)*A362278(k-3*j)*17^(j-1)*(3*j)!/(6^j*j!) for n >= 3.

%p a := exp(-x-3*x^2/2)*(1-exp(-17*x^3/6)) / (1+LambertW(-x)):

%p ser := series(a, x = 0, 21): seq(n!*coeff(ser, x, n), n = 0 .. 20);

%o (Python)

%o from functools import lru_cache

%o from math import comb, factorial as fl

%o @lru_cache(None)

%o def b(n): return 1 if n < 2 else b(n-1) - 3*(n-1)*b(n-2)

%o def a(n):

%o     return 17*sum(comb(n,k)*(-1)**(k+1)*(n-k)**(n-k)*

%o         sum(comb(k,3*j)*b(k-3*j)*17**(j-1)*fl(3*j)//(6**j*fl(j))

%o             for j in range(1,k//3+1))

%o         for k in range(n+1))

%o print([a(n) for n in range(21)])

%Y Column k=3 of triangle A347999.

%Y Cf. A000169, A001865, A209324, A350134, A362278, A394928.

%K nonn,easy

%O 0,4

%A _Mélika Tebni_, May 03 2026