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A245114 G.f. A(x) satisfies A(x)^3 = 1 + 9*x*A(x)^5.

%I A245114 #21 Sep 02 2025 04:06:46
%S A245114 1,3,36,585,10935,221697,4740120,105225318,2402040420,56029889025,
%T A245114 1329627118248,31998624800220,779102941714461,19157195459506230,
%U A245114 475034438632316400,11865382635213387504,298265217964573747095,7539795161286074350785,191548870595159091038640,4888023169106780049244275
%N A245114 G.f. A(x) satisfies A(x)^3 = 1 + 9*x*A(x)^5.
%H A245114 Robert Israel, <a href="/A245114/b245114.txt">Table of n, a(n) for n = 0..696</a>
%F A245114 a(n) = 9^n * binomial((5*n - 2)/3, n) / (2*n + 1).
%F A245114 2*a(n+3)*(n+3)*(n+2)*(n+1)*(2*n+7)=135*a(n)*(5*n+1)*(5*n+4)*(5*n+7)*(5*n+13). - _Robert Israel_, Jan 30 2018
%F A245114 G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^7). - _Seiichi Manyama_, Jun 20 2025
%F A245114 a(n) ~ 3^n * 5^(5*n/3-1/6) / (sqrt(Pi) * 2^(2*(n+2)/3) * n^(3/2)). - _Amiram Eldar_, Sep 02 2025
%e A245114 G.f.: A(x) = 1 + 3*x + 36*x^2 + 585*x^3 + 10935*x^4 + 221697*x^5 +...
%e A245114 where A(x)^3 = 1 + 9*x*A(x)^5:
%e A245114 A(x)^3 = 1 + 9*x + 135*x^2 + 2430*x^3 + 48195*x^4 + 1015740*x^5 +...
%e A245114 A(x)^5 = 1 + 15*x + 270*x^2 + 5355*x^3 + 112860*x^4 + 2480058*x^5 +...
%p A245114 rec:= 2*a(n+3)*(n+3)*(n+2)*(n+1)*(2*n+7)=135*a(n)*(5*n+1)*(5*n+4)*(5*n+7)*(5*n+13):
%p A245114 f:= gfun:-rectoproc({rec,a(0)=1,a(1)=3,a(2)=36},a(n),remember):
%p A245114 map(f, [$0..30]); # _Robert Israel_, Jan 30 2018
%t A245114 nmax = 19; sol = {a[0] -> 1};
%t A245114 Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[A[x]^3 - (1 + 9 x A[x]^5) + O[x]^(n + 1), x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax}];
%t A245114 sol /. HoldPattern[a[n_] -> k_] :> Set[a[n], k];
%t A245114 a /@ Range[0, nmax] (* _Jean-François Alcover_, Nov 01 2019 *)
%o A245114 (PARI) /* From A(x)^3 = 1 + 9*x*A(x)^5 : */
%o A245114 {a(n) = local(A=1+x);for(i=1,n,A=(1 + 9*x*A^5 +x*O(x^n))^(1/3));polcoeff(A,n)}
%o A245114 for(n=0,20,print1(a(n),", "))
%o A245114 (PARI) {a(n) = 9^n * binomial((5*n - 2)/3, n) / (2*n+1)}
%o A245114 for(n=0,20,print1(a(n),", "))
%Y A245114 Cf. A004987, A008931, A078532, A247029, A376282, A376636.
%Y A245114 Cf. A245112, A385119.
%K A245114 nonn,changed
%O A245114 0,2
%A A245114 _Paul D. Hanna_, Jul 31 2014