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A363314 Expansion of g.f. A(x) satisfying 1/4 = Sum_{n=-oo..+oo} x^n * (A(x) - x^n)^(n-1), with a(0) = 5.

%I A363314 #13 Jun 09 2023 21:26:28
%S A363314 5,32,496,9024,181296,3882848,86887712,2007577472,47530180736,
%T A363314 1147071160768,28114384217104,697913487791552,17511114852998912,
%U A363314 443374443981736160,11314170816869911232,290688529521060711424,7513202655833624201472,195216134898681278515232
%N A363314 Expansion of g.f. A(x) satisfying 1/4 = Sum_{n=-oo..+oo} x^n * (A(x) - x^n)^(n-1), with a(0) = 5.
%C A363314 a(n) == 0 (mod 4^2) for n > 0.
%H A363314 Paul D. Hanna, <a href="/A363314/b363314.txt">Table of n, a(n) for n = 0..200</a>
%F A363314 G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following.
%F A363314 (1) 1/4 = Sum_{n=-oo..+oo} x^n * (A(x) - x^n)^(n-1).
%F A363314 (2) 1/4 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n^2) / (1 - x^n*A(x))^(n+1).
%F A363314 (3) A(x)/4 = Sum_{n=-oo..+oo} x^(2*n) * (A(x) - x^n)^(n-1).
%F A363314 (4) A(x)/4 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n*(n-1)) / (1 - x^n*A(x))^(n+1).
%e A363314 G.f.: A(x) =  5 + 32*x + 496*x^2 + 9024*x^3 + 181296*x^4 + 3882848*x^5 + 86887712*x^6 + 2007577472*x^7 + 47530180736*x^8 + ...
%o A363314 (PARI) {a(n) = my(A=[5]); for(i=1,n, A = concat(A,0);
%o A363314 A[#A] = polcoeff(-4  + 4^2*sum(m=-#A, #A, x^m * (Ser(A) - x^m)^(m-1) ), #A-1););A[n+1]}
%o A363314 for(n=0,30,print1(a(n),", "))
%Y A363314 Cf. A357227, A363141, A363312, A363313, A363315.
%K A363314 nonn
%O A363314 0,1
%A A363314 _Paul D. Hanna_, May 28 2023