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%I #10 May 11 2019 13:03:40
%S 1,7,17,143,1297,10943,119041,1352319,16521601,217712895,3035672577,
%T 44699885311,692651630593,11245459802111,190749994213377,
%U 3372199652642815,61989222776496129,1182514506870886399,23367685697859391489,477573301465741901823,10079510247865215746049,219396394337370417070079,4918940850123829203173377,113466581251217062104399871,2690031991636202195545948161
%N G.f. A(x) satisfies: 1 = Sum_{n>=0} x^n * ((1 + 4*x)^n - A(x))^(n+1), where A(0) = 0.
%H Paul D. Hanna, <a href="/A325584/b325584.txt">Table of n, a(n) for n = 1..300</a>
%F G.f. A(x) satisfies:
%F (1) 1 = Sum_{n>=0} x^n * ((1 + 4*x)^n - A(x))^(n+1).
%F (2) 1 + x = Sum_{n>=0} x^n * (1 + 4*x)^(n*(n-1)) / (1 + x*(1 + 4*x)^n*A(x))^(n+1).
%F FORMULA FOR TERMS.
%F a(n) = (-1)^n (mod 4) for n >= 0.
%e G.f.: A(x) = x + 7*x^2 + 17*x^3 + 143*x^4 + 1297*x^5 + 10943*x^6 + 119041*x^7 + 1352319*x^8 + 16521601*x^9 + 217712895*x^10 + 3035672577*x^11 + ...
%e such that
%e 1 = (1 - A(x)) + x*((1+4*x) - A(x))^2 + x^2*((1+4*x)^2 - A(x))^3 + x^3*((1+4*x)^3 - A(x))^4 + x^4*((1+4*x)^4 - A(x))^5 + x^5*((1+4*x)^5 - A(x))^6 + x^6*((1+4*x)^6 - A(x))^7 + ...
%o (PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0); A[#A] = polcoeff( sum(m=0, #A, x^m*((1 + 4*x +x*O(x^#A))^m - x*Ser(A))^(m+1) ), #A); ); A[n+1]}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A307940, A325582, A325583, A325585.
%K nonn
%O 1,2
%A _Paul D. Hanna_, May 11 2019
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