cp's OEIS Frontend

A386415 G.f. A(x) satisfies A(x) = (1 + 9*x*A(x)^4)^(2/3).

%I A386415 #18 Aug 01 2025 08:56:34
%S A386415 1,6,135,4140,146475,5629338,228355281,9622693080,417122726490,
%T A386415 18480617374050,833136935399208,38094723501749460,1762459398803643930,
%U A386415 82353342267057244950,3880848811889775489300,184228926273804535479216,8801795826996054546077865,422898288144162288398536860
%N A386415 G.f. A(x) satisfies A(x) = (1 + 9*x*A(x)^4)^(2/3).
%H A386415 Paolo Xausa, <a href="/A386415/b386415.txt">Table of n, a(n) for n = 0..500</a>
%H A386415 Wikipedia, <a href="https://en.wikipedia.org/wiki/Fuss-Catalan_number">Fuss-Catalan number</a>
%F A386415 a(n) = 9^n * binomial((8*n+2)/3,n)/(4*n+1).
%F A386415 G.f.: B(x)^2, where B(x) is the g.f. of A386416.
%F A386415 D-finite with recurrence 5*n*(n-1)*(n-2)*(5*n-4)*(5*n+2)*(5*n-7)*(5*n-1)*a(n) -3456*(4*n-11)*(8*n-19)*(8*n-13)*(4*n-5)*(8*n-7)*(2*n-1)*(8*n-1)*a(n-3)=0. - _R. J. Mathar_, Jul 30 2025
%p A386415 A386415 := proc(n)
%p A386415     9^n*binomial((8*n+2)/3,n)/(4*n+1) ;
%p A386415 end proc:
%p A386415 seq(A386415(n),n=0..80) ; # _R. J. Mathar_, Jul 30 2025
%t A386415 A386415[n_] := 9^n * Binomial[(8*n + 2)/3, n]/(4*n + 1);
%t A386415 Array[A386415, 20, 0] (* _Paolo Xausa_, Aug 01 2025 *)
%o A386415 (PARI) apr(n, p, r) = r*binomial(n*p+r, n)/(n*p+r);
%o A386415 a(n) = 9^n*apr(n, 8/3, 2/3);
%Y A386415 Cf. A004989, A377269, A386413, A386414.
%Y A386415 Cf. A078532, A386416.
%K A386415 nonn,easy
%O A386415 0,2
%A A386415 _Seiichi Manyama_, Jul 21 2025