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A366732 Expansion of g.f. A(x) satisfying 0 = Sum_{n=-oo..+oo} x^n * A(x)^n * (2 - x^(n-1))^(n+1).

%I A366732 #6 Oct 29 2023 22:02:25
%S A366732 1,2,4,22,108,574,3224,18592,109728,660938,4041900,25034000,156724204,
%T A366732 990127086,6304425800,40416596578,260658078580,1689976752116,
%U A366732 11008752656960,72016455973262,472912945955364,3116243639293972,20599091568973324,136557058462319178,907668022344460584
%N A366732 Expansion of g.f. A(x) satisfying 0 = Sum_{n=-oo..+oo} x^n * A(x)^n * (2 - x^(n-1))^(n+1).
%C A366732 a(n) = Sum_{k=0..n} A366730(n,k) * 2^k for n >= 0.
%H A366732 Paul D. Hanna, <a href="/A366732/b366732.txt">Table of n, a(n) for n = 0..250</a>
%F A366732 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F A366732 (1) 0 = Sum_{n=-oo..+oo} x^n * A(x)^n * (2 - x^(n-1))^(n+1).
%F A366732 (2) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ( A(x)^n * (1 - 2*x^(n+1))^(n-1) ).
%e A366732 G.f.: A(x) = 1 + 2*x + 4*x^2 + 22*x^3 + 108*x^4 + 574*x^5 + 3224*x^6 + 18592*x^7 + 109728*x^8 + 660938*x^9 + 4041900*x^10 + 25034000*x^11 + ...
%o A366732 (PARI) {a(n) = my(A=[1]); for(i=1,n, A = concat(A,0);
%o A366732 A[#A] = polcoeff( sum(n=-#A,#A, x^n * Ser(A)^n * (2 - x^(n-1))^(n+1) ), #A-2));A[n+1]}
%o A366732 for(n=0,40,print1(a(n),", "))
%Y A366732 Cf. A366730, A366731, A366733, A366734, A366735.
%K A366732 nonn
%O A366732 0,2
%A A366732 _Paul D. Hanna_, Oct 29 2023