%I #7 Feb 03 2025 09:34:08
%S 0,1,1,1,2,11,51,177,519,1513,5042,18866,71270,257974,905425,3193737,
%T 11578842,42930441,159998493,593445318,2194106568,8138471667,
%U 30399156174,114219616809,430344635913,1622777285682,6125465491551
%N Number of rooted planar n-trees whose number of leaves is equal to 1 modulo 3.
%C a(n)+A119365(n)+A119367(n)=A000108(n).
%F a(n)=sum{k=0..n, if(mod(n-k,3)=1, (1/n)*C(n,k)*C(n,k+1), 0)}
%F a(0)=0, a(n)=sum{k=0..floor(n/3), (1/n)*C(n,3k+1)C(n,3k)},n>0; - _Paul Barry_, Jan 25 2007
%F Conjecture D-finite with recurrence +n*(881*n-4580)*(n-2)*(n+1)*a(n) -3*n*(612*n^3-2827*n^2-2988*n+10135)*a(n-1) +3*(-3088*n^4+42803*n^3-190361*n^2+313702*n-167988)*a(n-2) +(43042*n^4-600920*n^3+2924411*n^2-5860777*n+4115562)*a(n-3) +3*(-38600*n^4+558681*n^3-2904370*n^2+6389913*n-4965528)*a(n-4) +3*(-14776*n^4+162695*n^3-434711*n^2-415064*n+1878084)*a(n-5) -9*(n-6)*(10835*n^3-106831*n^2+290611*n-173519)*a(n-6) +54*(n-6)*(n-7)*(593*n-1429)*(2*n-13)*a(n-7)=0. - _R. J. Mathar_, Feb 03 2025
%p A119366 := proc(n)
%p if n = 0 then
%p 0;
%p else
%p add(binomial(n,3*k+1)*binomial(n,3*k),k=0..n/3) ;
%p %/n ;
%p end if;
%p end proc: # _R. J. Mathar_, Dec 02 2014
%K easy,nonn
%O 0,5
%A _Paul Barry_, May 16 2006