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A207643 a(n) = 1 + (n-1) + (n-1)*[n/2-1] + (n-1)*[n/2-1]*[n/3-1] + (n-1)*[n/2-1]*[n/3-1]*[n/4-1] +... for n>0 with a(0)=1, where [x] = floor(x).

%I A207643 #25 May 13 2013 01:50:04
%S A207643 1,1,2,3,7,9,26,31,71,129,262,291,1222,1333,2198,5139,11881,12673,
%T A207643 39594,41923,117326,251841,354292,371163,1870453,2598577,3456926,
%U A207643 7103955,16665859,17283113,72923314,75437911,165990152,335534913,422310802,695765699,3589651696
%N A207643 a(n) = 1 + (n-1) + (n-1)*[n/2-1] + (n-1)*[n/2-1]*[n/3-1] + (n-1)*[n/2-1]*[n/3-1]*[n/4-1] +... for n>0 with a(0)=1, where [x] = floor(x).
%C A207643 Radius of convergence of g.f. A(x) is near 0.54783..., with A(1/2) = 7.2672875151872...
%C A207643 Compare the definition of a(n) to the trivial binomial sum:
%C A207643 2^(n-1) = 1 + (n-1) + (n-1)*(n/2-1) + (n-1)*(n/2-1)*(n/3-1) + (n-1)*(n/2-1)*(n/3-1)*(n/4-1) +...
%H A207643 Paul D. Hanna, <a href="/A207643/b207643.txt">Table of n, a(n) for n = 0..500</a>
%F A207643 a(n) = 1 + Sum_{k=1..[n/2]} Product_{j=1..k} floor( (n-j) / j ).
%F A207643 Equals row sums of irregular triangle A207645.
%e A207643 a(2) = 1 + 1 = 2; a(3) = 1 + 2 = 3;
%e A207643 a(4) = 1 + 3 + 3*[4/2-1] = 7;
%e A207643 a(5) = 1 + 4 + 4*[5/2-1] = 9;
%e A207643 a(6) = 1 + 5 + 5*[6/2-1] + 5*[6/2-1]*[6/3-1] = 26;
%e A207643 a(7) = 1 + 6 + 6*[7/2-1] + 6*[7/2-1]*[7/3-1] = 31;
%e A207643 a(8) = 1 + 7 + 7*[8/2-1] + 7*[8/2-1]*[8/3-1] + 7*[8/2-1]*[8/3-1]*[8/4-1] = 71; ...
%t A207643 a[n_] := 1 + Sum[ Product[ Floor[(n-j)/j], {j, 1, k}], {k, 1, n/2}]; Table[a[n], {n, 0, 36}] (* _Jean-François Alcover_, Mar 06 2013 *)
%o A207643 (PARI) {a(n)=1+sum(k=1,n,prod(j=1,k,floor(n/j-1)))}
%o A207643 for(n=0,50,print1(a(n),", "))
%o A207643 (PARI) a(n)=my(t=1);1+sum(k=1,n,t*=n\k-1) \\ _Charles R Greathouse IV_, Feb 20 2012
%Y A207643 Cf. A075885, A207645, A207644, A207646, A207647.
%K A207643 nonn,nice
%O A207643 0,3
%A A207643 _Paul D. Hanna_, Feb 19 2012