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A120976 G.f. A(x) satisfies A(x/A(x)^5) = 1 + x ; thus A(x) = 1 + series_reversion(x/A(x)^5).

%I A120976 #9 Jun 04 2025 09:52:26
%S A120976 1,1,5,60,1060,23430,602001,17281760,541258390,18210836060,
%T A120976 651246905140,24566101401035,971933892729980,40156993344526860,
%U A120976 1726753293393763625,77065076699967844390,3561820706538663354320,170162336673835615653925,8389644485709060522744640
%N A120976 G.f. A(x) satisfies A(x/A(x)^5) = 1 + x ; thus A(x) = 1 + series_reversion(x/A(x)^5).
%F A120976 G.f. satisfies: A(x) = 1 + x*B(x)^5 = 1 + (1 + x*C(x)^5 )^5 where B(x) and C(x) satisfy: C(x) = B(x)*B(A(x)-1), B(x) = A(A(x)-1), B(A(x)-1) = A(B(x)-1), B(x/A(x)^5) = A(x), B(x) = A(x*B(x)^5) and B(x) is g.f. of A120977.
%F A120976 From _Seiichi Manyama_, Jun 04 2025: (Start)
%F A120976 Let b(n,k) = [x^n] B(x)^k, where B(x) is the g.f. of A120977.
%F A120976 b(n,0) = 0^n; b(n,k) = k * Sum_{j=0..n} binomial(5*n+k,j)/(5*n+k) * b(n-j,5*j).
%F A120976 a(n) = b(n-1,5) for n > 0. (End)
%o A120976 (PARI) {a(n)=local(A=[1,1]);for(i=2,n,A=concat(A,0); A[ #A]=-Vec(subst(Ser(A),x,x/Ser(A)^5))[ #A]);A[n+1]}
%o A120976 (PARI) b(n, k) = if(k==0, 0^n, k*sum(j=0, n, binomial(5*n+k, j)/(5*n+k)*b(n-j, 5*j)));
%o A120976 a(n) = if(n==0, 1, b(n-1, 5)); \\ _Seiichi Manyama_, Jun 04 2025
%Y A120976 Cf. A120977; variants: A120970, A120972, A120974, A030266, A067145, A107096.
%K A120976 nonn
%O A120976 0,3
%A A120976 _Paul D. Hanna_, Jul 20 2006