A376490 G.f. satisfies A(x) = 1 / (1 - x^3*A(x)^3 / (1 - x)).

1, 0, 0, 1, 1, 1, 5, 9, 13, 39, 87, 157, 389, 923, 1899, 4426, 10582, 23414, 54022, 128643, 295735, 686881, 1631513, 3825456, 8974024, 21330400, 50550032, 119644037, 285176865, 680215735, 1621245503, 3878312658, 9293056066, 22267588692, 53463982624

Offset: 0

Author

Seiichi Manyama, Sep 25 2024

Keywords

• nonn

Crossrefs

Cf. A002212, A376489, A376491.

Programs

Maple

A376490 := proc(n)
    add(binomial(4*k,k)*binomial(n-2*k-1,n-3*k)/(3*k+1),k=0..floor(n/3)) ;
end proc:
seq(A376490(n),n=0..70) ; # R. J. Mathar, Sep 26 2024

PARI

a(n) = sum(k=0, n\3, binomial(4*k, k)*binomial(n-2*k-1, n-3*k)/(3*k+1));

Formula

a(n) = Sum_{k=0..floor(n/3)} binomial(4*k,k) * binomial(n-2*k-1,n-3*k) / (3*k+1).

D-finite with recurrence 243*n*(n-1)*(n+1)*a(n) -81*n*(n-1)*(16*n-29)*a(n-1) +27*(106*n-285)*(n-1)*(n-2)*a(n-2) +9*(-628*n^3+4365*n^2-10585*n+8778)*a(n-3) +3*(4057*n^3-33849*n^2+94446*n-89368)*a(n-4) +2*(-8954*n^3+98325*n^2-354169*n+419010)*a(n-5) +12*(1225*n^3-17314*n^2+80552*n-123168)*a(n-6) -384*(2*n-13)*(8*n^2-88*n+239)*a(n-7) +256*(2*n-15)*(n-7)*(2*n-13)*a(n-8)=0. - R. J. Mathar, Sep 26 2024