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%I #32 Mar 09 2018 21:03:10
%S 1,1,3,18,120,1170,12600,176400,2608200,46607400,883159200,
%T 19429502400,447567120000,11629447830000,316028116404000,
%U 9516436753824000,297478346845680000,10151626256147376000,359237701318479984000,13733349319337487840000,542212802070902202240000
%N Sequence defined by a recurrence close to that of A001147
%C From _Christopher J. Smyth_, Jan 26 2018: (Start)
%C The sequence is defined by the recurrence formula below. This recurrence is very similar to that of the sequence b(n) = A001147(n-1), which satisfies b(1)=1 and, for n >= 2, b(n) = Sum_{i=1..floor((n-1)/2)} binomial(n, i) * b(i) * b(n-i) + B, where B = 0 (n odd), = (1/2)*binomial(n, n/2)*b(n/2)^2 (n even) [see formula of Walsh on A001147 page]. Removal of the factor 1/2 from the definition of B gives, for n >= 3, the formula below for a(n).
%C This sequence seems to have been defined in the mistaken belief that it had applications. In fact the applications stated on earlier versions of this page actually belonged to A001147 -- see my comment on the A001147 page.
%C (End)
%H Michael De Vlieger, <a href="/A109714/b109714.txt">Table of n, a(n) for n = 1..403</a>
%F a(1) = 1, a(2) = 1 and a(n) = Sum_{i=1..floor(n/2)} binomial(n, i) * a(i) * a(n-i) for n >= 3.
%e a(3) = 3*a(1)*a(2) = 3, a(4) = 4*a(1)*a(3) + 6*a(2)^2 = 18.
%t Fold[Append[#1, Sum[Binomial[#2, i] #1[[i]] #1[[#2 - i]], {i, Floor[#2/2]}]] &, {1, 1}, Range[3, 21]] (* _Michael De Vlieger_, Dec 13 2017 *)
%o (MATLAB) function m = a(n); if n==1 m = 1; elseif n==2 m = 1; else m = 0; for i=1:floor(n/2); f1 = binomial(n,i); f2 = a(i); f3 = a(n-i); m = m + f1*f2*f3; end; end;
%Y Cf. A001147
%K easy,nonn
%O 1,3
%A Niko Brummer (niko.brummer(AT)gmail.com), Aug 08 2005
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