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G.f. A(x) satisfies: 1 - x = Sum_{n>=0} (x^(3*n) + (-1)^n*A(x))^n.
3

%I #5 Apr 07 2022 12:11:25

%S 1,1,2,3,4,6,11,19,33,60,110,201,372,696,1305,2456,4654,8855,16896,

%T 32366,62217,119910,231704,448879,871531,1695541,3305057,6454033,

%U 12624041,24731009,48520396,95324187,187517959,369329721,728262059,1437591201,2840751293

%N G.f. A(x) satisfies: 1 - x = Sum_{n>=0} (x^(3*n) + (-1)^n*A(x))^n.

%F G.f. A(x) satisfies:

%F (1) 1 - x = Sum_{n>=0} ( x^(3*n) + (-1)^n*A(x) )^n.

%F (2) 1 - x = Sum_{n>=0} x^(3*n^2) / (1 + (-1)^n*x^(3*n)*A(x))^(n+1).

%e G.f.: A(x) = x + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 6*x^6 + 11*x^7 + 19*x^8 + 33*x^9 + 60*x^10 + 110*x^11 + 201*x^12 + 372*x^13 + 696*x^14 + ...

%e where

%e 1 - x = 1 + (x^3 - A(x)) + (x^6 + A(x))^2 + (x^9 - A(x))^3 + (x^12 + A(x))^4 + (x^15 - A(x))^5 + (x^18 + A(x))^6 + ...

%e Also,

%e 1 - x = 1/(1 + A(x)) + x^3/(1 - x^3*A(x))^2 + x^12/(1 + x^6*A(x))^3 + x^27/(1 - x^9*A(x))^4 + x^48/(1 + x^12*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^(3*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\3), x^(3*m^2)/(1 + (-x)^(3*m)*x*Ser(A))^(m+1) ),#A));A[n+1]}

%o for(n=0,40,print1(a(n),", "))

%Y Cf. A317997, A352818, A352820, A352821.

%K nonn

%O 1,3

%A _Paul D. Hanna_, Apr 05 2022