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A109714
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Sequence defined by a recurrence close to that of A001147
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1
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1, 1, 3, 18, 120, 1170, 12600, 176400, 2608200, 46607400, 883159200, 19429502400, 447567120000, 11629447830000, 316028116404000, 9516436753824000, 297478346845680000, 10151626256147376000, 359237701318479984000, 13733349319337487840000, 542212802070902202240000
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OFFSET
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1,3
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COMMENTS
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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).
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.
(End)
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LINKS
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FORMULA
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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.
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EXAMPLE
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a(3) = 3*a(1)*a(2) = 3, a(4) = 4*a(1)*a(3) + 6*a(2)^2 = 18.
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MATHEMATICA
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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 *)
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PROG
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(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;
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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Niko Brummer (niko.brummer(AT)gmail.com), Aug 08 2005
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STATUS
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approved
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