A006675 - OEIS

%I M2079 #23 Dec 18 2021 22:20:31

%S 0,0,2,15,104,770,6264,56196,554112,5973264,69998400,886897440,

%T 12089295360,176484597120,2748022986240,45472329504000,

%U 796983880089600,14751208762214400,287543058350284800

%N Number of paths through an array.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A006675/b006675.txt">Table of n, a(n) for n=0..100</a>

%H A. M. Khidr and B. S. El-Desouky, <a href="http://dx.doi.org/10.1016/S0195-6698(84)80018-9">A symmetric sum involving the Stirling numbers of the first kind</a>, European J. Combin., 5 (1984), 51-54.

%F a(n) = n*n! * (H(n) - 1) where H(n) = Sum_{k=1..n} 1/k.

%F E.g.f. A(x) = x*(1-x)^-2 * (-log(1-x)).

%F a(n) = A001705(n) - A000254(n). - _Peter Bala_, Feb 12 2019

%e x*(1-x)^-2 * (-log(1-x)) = x^2 + (5/2)*x^3 + (13/3)*x^4 + (77/12)*x^5 + ...

%t a[n_] := n*n!*(HarmonicNumber[n]-1); Table[a[n], {n, 0, 18}] (* _Jean-François Alcover_, Nov 28 2011 *)

%Y Cf. A000254, A001705.

%K nonn,easy,nice

%O 0,3

%A _N. J. A. Sloane_

%E More terms from Joe Keane (jgk(AT)jgk.org)