A386380 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/6)} a(6*k) * a(n-1-6*k).

1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 15, 24, 34, 45, 57, 70, 154, 253, 368, 500, 650, 819, 1827, 3045, 4495, 6200, 8184, 10472, 23562, 39627, 59052, 82251, 109668, 141778, 320866, 543004, 814506, 1142295, 1533939, 1997688, 4540200, 7718340, 11633440, 16398200, 22137570

Offset: 0

Author

Seiichi Manyama, Jul 20 2025

Keywords

• nonn

Links

• Wikipedia, Fuss-Catalan number

Crossrefs

Cf. A002296, A233832, A233833, A386392, A233834, A130565.

Cf. A047749, A118968, A124753, A386379, A386396.

Programs

Maple

A386380 := proc(n)
    option remember ;
    if n = 0 then
        1;
    else
        add(procname(6*k)*procname(n-1-6*k),k=0..floor((n-1)/6)) ;
    end if;
end proc:
seq(A386380(n),n=0..80) ; # R. J. Mathar, Jul 30 2025

PARI

apr(n, p, r) = r*binomial(n*p+r, n)/(n*p+r);
a(n) = apr(n\6, 7, n%6+1);

Formula

For k=0..5, a(6*n+k) = (k+1) * binomial(7*n+k+1,n)/(7*n+k+1).

G.f. A(x) satisfies A(x) = 1/(1 - x * Product_{k=0..5} A(w^k*x)), where w = exp(Pi*i/3).