A372003 G.f. A(x) satisfies A(x) = ( 1 + 9*x*(1 + x*A(x)) )^(1/3).

1, 3, -6, 36, -216, 1404, -9648, 68904, -506304, 3802464, -29055024, 225142416, -1764900576, 13970400480, -111506362560, 896391836928, -7251109424640, 58978357310592, -482049643011840, 3957079727715840, -32609916223598592, 269682253882186752

Offset: 0

Author

Seiichi Manyama, Apr 15 2024

Keywords

• sign

Crossrefs

Cf. A372002.

Programs

Maple

A372003 := proc(n)
    add(9^k*binomial((n-k+1)/3,k)*binomial(k,n-k)/(n-k+1),k=0..n) ;
end proc:
seq(A372003(n),n=0..60) ; # R. J. Mathar, Apr 22 2024

PARI

a(n) = sum(k=0, n, 9^k*binomial(n/3-k/3+1/3, k)*binomial(k, n-k)/(n-k+1));

Formula

a(n) = Sum_{k=0..n} 9^k * binomial(n/3-k/3+1/3,k) * binomial(k,n-k)/(n-k+1).

D-finite with recurrence n*(n-2)*a(n) +3*(8*n^2-31*n+24)*a(n-1) +27*(7*n^2-41*n+56)*a(n-2) +54*(3*n-10)*(3*n-14)*a(n-3) -108*(n-4)*(n-7)*a(n-6) -648*(n-6)*(n-8)*a(n-7)=0. - R. J. Mathar, Apr 22 2024