A119366 Number of rooted planar n-trees whose number of leaves is equal to 1 modulo 3.

0, 1, 1, 1, 2, 11, 51, 177, 519, 1513, 5042, 18866, 71270, 257974, 905425, 3193737, 11578842, 42930441, 159998493, 593445318, 2194106568, 8138471667, 30399156174, 114219616809, 430344635913, 1622777285682, 6125465491551

Offset: 0

Author

Paul Barry, May 16 2006

Keywords

• easy
• nonn

Comments

a(n)+A119365(n)+A119367(n)=A000108(n).

Programs

Maple

A119366 := proc(n)
    if n = 0 then
        0;
    else
        add(binomial(n,3*k+1)*binomial(n,3*k),k=0..n/3) ;
        %/n ;
    end if;
end proc: # R. J. Mathar, Dec 02 2014

Formula

a(n)=sum{k=0..n, if(mod(n-k,3)=1, (1/n)*C(n,k)*C(n,k+1), 0)}

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

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