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a(n) = (Sum_{j=1..(K-1)/2} a(n-2*j+1)*a(n-2*j))/a(n-K) with a(1),...,a(K)=1, where K=7.
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%I #12 Mar 17 2017 22:02:16

%S 1,1,1,1,1,1,1,3,5,17,89,1529,136169,208203929,9450306981389,

%T 393518208756305576259,218756933853729911666129962011049,

%U 967245356889229414311807903425793640476269370763657,138385630187929719100558260443148868172248984484844599503721372030874017578780705

%N a(n) = (Sum_{j=1..(K-1)/2} a(n-2*j+1)*a(n-2*j))/a(n-K) with a(1),...,a(K)=1, where K=7.

%H Seiichi Manyama, <a href="/A283331/b283331.txt">Table of n, a(n) for n = 1..24</a>

%H Matthew Christopher Russell, <a href="http://www.math.rutgers.edu/~zeilberg/Theses/MatthewRussellThesis.pdf">Using experimental mathematics to conjecture and prove theorems in the theory of partitions and commutative and non-commutative recurrences</a>, PhD Dissertation, Mathematics Department, Rutgers University, May 2016; see <a href="https://pdfs.semanticscholar.org/fdeb/e20954dacb7ec7a24afe2cf491b951c5a28d.pdf">also</a>.

%t a[n_]:=If[n<8, 1, Sum[a[n - 2j + 1]*a[n-2j], {j, 3}]/a[n - 7]]; Table[a[n], {n, 1, 20}] (* _Indranil Ghosh_, Mar 17 2017 *)

%o (PARI) a(n)= if(n<8, 1, sum(j=1, 3, a(n - 2*j + 1)*a(n - 2*j))/a(n - 7));

%o for(n=1, 20, print1(a(n),", ")) \\ _Indranil Ghosh_, Mar 17 2017

%K nonn

%O 1,8

%A _N. J. A. Sloane_, Mar 17 2017

%E More terms from _Indranil Ghosh_, Mar 17 2017