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A363105 Expansion of g.f. A(x) satisfying 5 = Sum_{n=-oo..+oo} (-x)^n * (5*A(x) + x^(n-1))^(n+1).

%I A363105 #7 May 22 2023 08:57:22
%S A363105 1,7,59,538,5149,51059,520035,5407889,57181230,612910369,6644662132,
%T A363105 72731584789,802696690614,8922392225233,99798739026795,
%U A363105 1122441028044882,12686176392341722,144013323190860339,1641303449002365323,18772674107796041770,215413772477355781876
%N A363105 Expansion of g.f. A(x) satisfying 5 = Sum_{n=-oo..+oo} (-x)^n * (5*A(x) + x^(n-1))^(n+1).
%H A363105 Paul D. Hanna, <a href="/A363105/b363105.txt">Table of n, a(n) for n = 0..300</a>
%F A363105 G.f. A(x) = Sum_{n>=0} a(n) * x^n may be described as follows.
%F A363105 (1) 5 = Sum_{n=-oo..+oo} (-1)^n * x^n * (5*A(x) + x^(n-1))^(n+1).
%F A363105 (2) 5 = Sum_{n=-oo..+oo} (-1)^n * x^(3*n+1) * (5*A(x) + x^n)^n.
%F A363105 (3) 5*x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + 5*A(x)*x^(n+1))^(n-1).
%F A363105 (4) 5*x = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n*(n-1)) / (1 + 5*A(x)*x^(n+1))^(n+1).
%F A363105 (5) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^n * x^n * (5*A(x) + x^(n-1))^n ].
%F A363105 (6) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^(n+1) * x^(2*n+1) * (5*A(x) + x^n)^n ].
%F A363105 (7) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 5*A(x)*x^(n+1))^n ].
%F A363105 (8) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (5*A(x) + x^n)^(n+1).
%F A363105 (9) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + 5*A(x)*x^n)^n.
%F A363105 (10) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + 5*A(x)*x^(n+1))^n.
%F A363105 a(n) = Sum_{k=0..n} A359670(n,k) * 5^k for n >= 0.
%e A363105 G.f.: A(x) = 1 + 7*x + 59*x^2 + 538*x^3 + 5149*x^4 + 51059*x^5 + 520035*x^6 + 5407889*x^7 + 57181230*x^8 + 612910369*x^9 + 6644662132*x^10 + ...
%o A363105 (PARI) {a(n) = my(A=1, y=5); for(i=1, n,
%o A363105 A = 1/sum(m=-#A, #A, (-1)^m * (x*y*A + x^m + x*O(x^n) )^m ) );
%o A363105 polcoeff( A, n, x)}
%o A363105 for(n=0, 25, print1( a(n), ", "))
%o A363105 (PARI) {a(n) = my(A=[1], y=5); for(i=1, n, A = concat(A, 0);
%o A363105 A[#A] = polcoeff(-y + sum(n=-#A, #A, (-1)^n * x^n * (y*Ser(A) + x^(n-1))^(n+1) )/(-y), #A-1, x) ); A[n+1]}
%o A363105 for(n=0, 25, print1( a(n), ", "))
%Y A363105 Cf. A359670, A359711, A359712, A359713, A363104.
%Y A363105 Cf. A363185.
%K A363105 nonn
%O A363105 0,2
%A A363105 _Paul D. Hanna_, May 21 2023