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 A006675 Number of paths through an array. (Formerly M2079) 8

%I M2079

%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), H[ n ] = sum[ k=1..n ] k^-1.

%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)

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Last modified October 16 20:55 EDT 2019. Contains 328103 sequences. (Running on oeis4.)