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

0,5

Comments

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

Links

Table of n, a(n) for n=0..26.

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

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

Crossrefs

Sequence in context: A018933 A375427 A116586 * A034574 A054665 A134963

Adjacent sequences: A119363 A119364 A119365 * A119367 A119368 A119369

Keyword

easy,nonn

Author

Paul Barry, May 16 2006

Status

approved