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%I #5 Apr 07 2022 12:11:51
%S 1,1,1,1,2,3,4,5,6,8,13,21,32,46,66,99,155,244,376,569,862,1328,2077,
%T 3256,5071,7853,12181,19023,29882,47004,73808,115757,181785,286323,
%U 452111,714548,1129185,1784586,2823069,4472449,7094472,11261549,17882350
%N G.f. A(x) satisfies: 1 - x = Sum_{n>=0} (x^(5*n) + (-1)^n*A(x))^n.
%F G.f. A(x) satisfies:
%F (1) 1 - x = Sum_{n>=0} ( x^(5*n) + (-1)^n*A(x) )^n.
%F (2) 1 - x = Sum_{n>=0} x^(5*n^2) / (1 + (-1)^n*x^(5*n)*A(x))^(n+1).
%e G.f.: A(x) = x + x^2 + x^3 + x^4 + 2*x^5 + 3*x^6 + 4*x^7 + 5*x^8 + 6*x^9 + 8*x^10 + 13*x^11 + 21*x^12 + 32*x^13 + 46*x^14 + 66*x^15 + ...
%e where
%e 1 - x = 1 + (x^5 - A(x)) + (x^10 + A(x))^2 + (x^15 - A(x))^3 + (x^20 + A(x))^4 + (x^25 - A(x))^5 + (x^30 + A(x))^6 + ...
%e Also,
%e 1 - x = 1/(1 + A(x)) + x^5/(1 - x^5*A(x))^2 + x^20/(1 + x^10*A(x))^3 + x^45/(1 - x^15*A(x))^4 + x^80/(1 + x^20*A(x))^5 + ...
%o (PARI) {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
%o A[#A] = polcoeff( sum(m=0,#A, (x^(5*m) + (-1)^m*x*Ser(A))^m ),#A));A[n+1]}
%o for(n=0,40,print1(a(n),", "))
%o (PARI) {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
%o A[#A] = polcoeff( sum(m=0,sqrtint(#A\5), x^(5*m^2)/(1 + (-x)^(5*m)*x*Ser(A))^(m+1) ),#A));A[n+1]}
%o for(n=0,40,print1(a(n),", "))
%Y Cf. A317997, A352818, A352819, A352820.
%K nonn
%O 1,5
%A _Paul D. Hanna_, Apr 05 2022
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