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A366735 Expansion of g.f. A(x) satisfying 0 = Sum_{n=-oo..+oo} x^n * A(x)^n * (1 + (-x)^(n-1))^(n+1).

%I A366735 #7 Oct 29 2023 22:02:49
%S A366735 1,1,4,14,54,218,911,3917,17235,77251,351498,1619362,7538944,35412306,
%T A366735 167626988,798823025,3829325596,18453005188,89338777895,434343634600,
%U A366735 2119679152092,10379998771157,50989711920778,251194614740028,1240735313801625,6143268099066535
%N A366735 Expansion of g.f. A(x) satisfying 0 = Sum_{n=-oo..+oo} x^n * A(x)^n * (1 + (-x)^(n-1))^(n+1).
%C A366735 a(n) = (-1)^n * Sum_{k=0..n} A366730(n,k) * (-1)^k for n >= 0.
%H A366735 Paul D. Hanna, <a href="/A366735/b366735.txt">Table of n, a(n) for n = 0..300</a>
%F A366735 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F A366735 (1) 0 = Sum_{n=-oo..+oo} x^n * A(x)^n * (1 + (-x)^(n-1))^(n+1).
%F A366735 (2) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ( A(x)^n * (1 + (-x)^(n+1))^(n-1) ).
%e A366735 G.f.: A(x) = 1 + x + 4*x^2 + 14*x^3 + 54*x^4 + 218*x^5 + 911*x^6 + 3917*x^7 + 17235*x^8 + 77251*x^9 + 351498*x^10 + 1619362*x^11 + 7538944*x^12 + ...
%o A366735 (PARI) {a(n) = my(A=[1]); for(i=1,n, A = concat(A,0);
%o A366735 A[#A] = polcoeff( sum(n=-#A,#A, x^n * Ser(A)^n * (1 + (-x)^(n-1))^(n+1) ), #A-2));H=A;A[n+1]}
%o A366735 for(n=0,30,print1(a(n),", "))
%Y A366735 Cf. A366730, A366731, A366732, A366733, A366734.
%K A366735 nonn
%O A366735 0,3
%A A366735 _Paul D. Hanna_, Oct 29 2023