cp's OEIS Frontend

A109714 Sequence defined by a recurrence close to that of A001147.

%I A109714 #34 Jun 02 2025 00:28:13
%S A109714 1,1,3,18,120,1170,12600,176400,2608200,46607400,883159200,
%T A109714 19429502400,447567120000,11629447830000,316028116404000,
%U A109714 9516436753824000,297478346845680000,10151626256147376000,359237701318479984000,13733349319337487840000,542212802070902202240000
%N A109714 Sequence defined by a recurrence close to that of A001147.
%C A109714 From _Christopher J. Smyth_, Jan 26 2018: (Start)
%C A109714 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 A109714 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 A109714 (End)
%H A109714 Michael De Vlieger, <a href="/A109714/b109714.txt">Table of n, a(n) for n = 1..403</a>
%F A109714 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 A109714 a(3) = 3*a(1)*a(2) = 3, a(4) = 4*a(1)*a(3) + 6*a(2)^2 = 18.
%t A109714 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 A109714 (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 A109714 Cf. A001147
%K A109714 easy,nonn
%O A109714 1,3
%A A109714 Niko Brummer (niko.brummer(AT)gmail.com), Aug 08 2005