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

%I A386380 #25 Jul 30 2025 04:49:44
%S A386380 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,
%T A386380 1827,3045,4495,6200,8184,10472,23562,39627,59052,82251,109668,141778,
%U A386380 320866,543004,814506,1142295,1533939,1997688,4540200,7718340,11633440,16398200,22137570
%N A386380 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/6)} a(6*k) * a(n-1-6*k).
%H A386380 Wikipedia, <a href="https://en.wikipedia.org/wiki/Fuss-Catalan_number">Fuss-Catalan number</a>
%F A386380 For k=0..5, a(6*n+k) = (k+1) * binomial(7*n+k+1,n)/(7*n+k+1).
%F A386380 G.f. A(x) satisfies A(x) = 1/(1 - x * Product_{k=0..5} A(w^k*x)), where w = exp(Pi*i/3).
%p A386380 A386380 := proc(n)
%p A386380     option remember ;
%p A386380     if n = 0 then
%p A386380         1;
%p A386380     else
%p A386380         add(procname(6*k)*procname(n-1-6*k),k=0..floor((n-1)/6)) ;
%p A386380     end if;
%p A386380 end proc:
%p A386380 seq(A386380(n),n=0..80) ; # _R. J. Mathar_, Jul 30 2025
%o A386380 (PARI) apr(n, p, r) = r*binomial(n*p+r, n)/(n*p+r);
%o A386380 a(n) = apr(n\6, 7, n%6+1);
%Y A386380 Cf. A002296, A233832, A233833, A386392, A233834, A130565.
%Y A386380 Cf. A047749, A118968, A124753, A386379, A386396.
%K A386380 nonn
%O A386380 0,8
%A A386380 _Seiichi Manyama_, Jul 20 2025