A386396 - OEIS

%I #14 Jul 20 2025 15:02:02

%S 1,1,1,1,1,1,1,1,2,3,4,5,6,7,8,17,27,38,50,63,77,92,200,325,468,630,

%T 812,1015,1240,2728,4488,6545,8925,11655,14763,18278,40508,67158,

%U 98728,135751,178794,228459,285384,635628,1059380,1566040,2165800,2869685,3689595,4638348

%N a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/7)} a(7*k) * a(n-1-7*k).

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Fuss-Catalan_number">Fuss-Catalan number</a>

%F For k=0..6, a(7*n+k) = (k+1) * binomial(8*n+k+1,n)/(8*n+k+1).

%F G.f. A(x) satisfies A(x) = 1/(1 - x * Product_{k=0..6} A(w^k*x)), where w = exp(2*Pi*i/7).

%o (PARI) apr(n, p, r) = r*binomial(n*p+r, n)/(n*p+r);

%o a(n) = apr(n\7, 8, n%7+1);

%Y Cf. A007556, A234461, A234462, A234463, A234464, A234465, A234466.

%Y Cf. A047749, A118968, A124753, A386379, A386380.

%K nonn

%O 0,9

%A _Seiichi Manyama_, Jul 20 2025