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%I #6 Nov 21 2013 13:11:31
%S 1,2,3,6,45,103285,637700839095606788040,
%T 47907611227303520484704817869777341656612683981478793109229998610027375657813231974364873146104781203691314770
%N Iterated binomial coefficients.
%C Defining a partial multiplication by nm = "n choose m" (where n+1>m) this is simply (...((((n)(n-1))(n-2))(n-3)...)3)2)1. With the brackets ordered in the opposite direction, as in: n((n-1)((n-2)....(3(2(1)))...)), is obviously simply n.
%F a(n) = (...(((n choose n-1) choose n-2) choose n-3)... choose 2) choose 1
%e E.g. a(5)=45 because 5C4=5 and then 5C3=10 and then 10C2=45 and finally 45C1=45.
%t b[n_, k_] := Binomial[n, k]; b[n_, k_] /; k == n-1 := n ; b[n_, k_] /; k == n-2 := n(n-1)/2; b[n_, k_] /; k == n-3 := n(n-1)(n-2)/6; a[n_] := Fold[b[#1, #1 - #2] &, n, Range[n-1, 1, -1]]; Table[a[n], {n, 1, 8}] (* _Jean-François Alcover_, Dec 16 2011 *)
%Y Cf. A000142.
%K nonn,nice,easy
%O 1,2
%A Marcel Jackson (Marcel.Jackson(AT)utas.edu.au)
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