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A363139 Expansion of A(x) satisfying -x = Sum_{n=-oo..+oo} (-x)^n * (1 - (-x)^n)^n / A(x)^n.

%I A363139 #9 May 31 2023 09:07:35
%S A363139 1,1,2,3,10,29,72,190,520,1413,3888,10839,30421,86218,246499,708931,
%T A363139 2050584,5962100,17407554,51019081,150052163,442677295,1309668356,
%U A363139 3884884796,11551622175,34425468793,102807253860,307617338332,922112808168,2768808168311,8327028966970
%N A363139 Expansion of A(x) satisfying -x = Sum_{n=-oo..+oo} (-x)^n * (1 - (-x)^n)^n / A(x)^n.
%C A363139 Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^n)^n, which holds formally for all y.
%H A363139 Paul D. Hanna, <a href="/A363139/b363139.txt">Table of n, a(n) for n = 0..400</a>
%F A363139 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following.
%F A363139 (1) -x = Sum_{n=-oo..+oo} (-x)^n * (1 - (-x)^n)^n / A(x)^n.
%F A363139 (2) -x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (1 - (-x)^n)^n.
%e A363139 G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 10*x^4 + 29*x^5 + 72*x^6 + 190*x^7 + 520*x^8 + 1413*x^9 + 3888*x^10 + 10839*x^11 + 30421*x^12 + ...
%e A363139 SPECIFIC VALUES.
%e A363139 G.f. A(x) diverges at x = 1/3.
%e A363139 A(1/sqrt(10)) = 2.740968311596221258712215041101550216...
%e A363139 A(3/10) = 2.04409403049365965943794935957987166879615299154...
%e A363139 A(x) = 2 at x = 0.29764678443183662600376771573865711430158997980267844885...
%e A363139 A(1/4) = 1.54451964019778087973376938515481313055726531377...
%o A363139 (PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);
%o A363139 A[#A] = polcoeff(x + sum(n=-#A, #A, (-x)^n * (1 - (-x)^n +x*O(x^#A))^n / Ser(A)^n ), #A-1) ); A[n+1]}
%o A363139 for(n=0, 30, print1(a(n), ", "))
%Y A363139 Cf. A357399.
%K A363139 nonn
%O A363139 0,3
%A A363139 _Paul D. Hanna_, May 30 2023