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A143501 G.f. A(x) satisfies A(x) = 1 + x*A(x*A(x)^3).

%I A143501 #9 Jun 04 2025 09:52:58
%S A143501 1,1,1,4,16,92,616,4729,40776,388057,4028230,45207583,544680014,
%T A143501 7004865885,95694153485,1382946630490,21067128029388,337224872043659,
%U A143501 5656357906530796,99168643108816180,1813250965008114981,34506927801196386939,682210688659107549765,13988252589131863518730
%N A143501 G.f. A(x) satisfies A(x) = 1 + x*A(x*A(x)^3).
%F A143501 G.f. satisfies: G(x) = x/[1 + A(x)*G(x)]^3 = x/A(G(x))^3 where G(x*A(x)^3) = x.
%F A143501 From _Seiichi Manyama_, Jun 04 2025: (Start)
%F A143501 Let a(n,k) = [x^n] A(x)^k.
%F A143501 a(n,0) = 0^n; a(n,k) = k * Sum_{j=0..n} binomial(3*n-3*j+k,j)/(3*n-3*j+k) * a(n-j,j). (End)
%e A143501 G.f. A(x) = 1 + x + x^2 + 4*x^3 + 16*x^4 + 92*x^5 + 616*x^6 + 4729*x^7 +...
%e A143501 A(x)^3 = 1 + 3*x + 6*x^2 + 19*x^3 + 78*x^4 + 411*x^5 + 2617*x^6 +...
%e A143501 A(x*A(x)^3) = 1 + x + 4*x^2 + 16*x^3 + 92*x^4 + 616*x^5 + 4729*x^6 +...
%e A143501 If G(x*A(x)^3) = x then
%e A143501 G(x) = x - 3*x^2 + 12*x^3 - 64*x^4 + 372*x^5 - 2385*x^6 + 15675*x^7 -+...
%e A143501 A(G(x)) = 1 + A(x)*G(x) = (x/G(x))^(1/3) where
%e A143501 A(x)*G(x) = x - 2*x^2 + 10*x^3 - 51*x^4 + 324*x^5 - 1985*x^6 + 13938*x^7 -...
%o A143501 (PARI) {a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=1+x*subst(A,x,x*A^3));polcoeff(A,n)}
%o A143501 (PARI) a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(3*n-3*j+k, j)/(3*n-3*j+k)*a(n-j, j))); \\ _Seiichi Manyama_, Jun 04 2025
%Y A143501 Cf. A087949, A143500.
%K A143501 nonn
%O A143501 0,4
%A A143501 _Paul D. Hanna_, Aug 20 2008