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A143437 G.f. A(x) satisfies A(x) = 1 + x*A(x*A(x))^5.

%I A143437 #7 Jun 05 2025 09:52:18
%S A143437 1,1,5,40,405,4745,61551,862050,12831835,200874055,3282575310,
%T A143437 55693595381,977058059380,17668078651755,328497282637520,
%U A143437 6267311264123850,122498870023756800,2449635783413544555,50061311067746399725,1044531750427750075150,22233430278290842445120
%N A143437 G.f. A(x) satisfies A(x) = 1 + x*A(x*A(x))^5.
%F A143437 G.f. satisfies: x - G(x) = G(x)^2*A(x)^5 where G(x*A(x)) = x.
%F A143437 From _Seiichi Manyama_, Jun 05 2025: (Start)
%F A143437 Let a(n,k) = [x^n] A(x)^k.
%F A143437 a(n,0) = 0^n; a(n,k) = k * Sum_{j=0..n} binomial(n-j+k,j)/(n-j+k) * a(n-j,5*j). (End)
%e A143437 G.f.: A(x) = 1 + x + 5*x^2 + 40*x^3 + 405*x^4 + 4745*x^5 + 61551*x^6 +...
%e A143437 A(x*A(x)) = 1 + x + 6*x^2 + 55*x^3 + 620*x^4 + 7940*x^5 + 111166*x^6 +...
%e A143437 A(x*A(x))^5 = 1 + 5*x + 40*x^2 + 405*x^3 + 4745*x^4 + 61551*x^5 +...
%o A143437 (PARI) {a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=1+x*subst(A^5,x,x*A));polcoeff(A,n)}
%o A143437 (PARI) a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(n-j+k, j)/(n-j+k)*a(n-j, 5*j))); \\ _Seiichi Manyama_, Jun 05 2025
%Y A143437 Cf. A143426, A143435, A143436.
%K A143437 nonn
%O A143437 0,3
%A A143437 _Paul D. Hanna_, Aug 14 2008