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A357399 Coefficients of x^n, n >= 0, in A(x) such that: x = Sum_{n=-oo..+oo} (-x)^n * (1 - (-x)^n)^n * A(x)^n.

%I A357399 #17 Dec 24 2022 13:51:41
%S A357399 1,1,3,10,37,143,564,2270,9305,38755,163569,698186,3009129,13077850,
%T A357399 57250728,252221229,1117409653,4975095073,22249463540,99901607730,
%U A357399 450187852401,2035353779794,9229671434155,41968536407303,191318458136066,874179701912764,4002949886221529
%N A357399 Coefficients of x^n, n >= 0, in A(x) such that: x = Sum_{n=-oo..+oo} (-x)^n * (1 - (-x)^n)^n * A(x)^n.
%C A357399 Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^n)^n, which holds formally for all y.
%H A357399 Paul D. Hanna, <a href="/A357399/b357399.txt">Table of n, a(n) for n = 0..300</a>
%F A357399 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following.
%F A357399 (1) x = Sum_{n=-oo..+oo} (-x)^n * (1 - (-x)^n)^n * A(x)^n.
%F A357399 (2) x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ((1 - (-x)^n)^n * A(x)^n).
%e A357399 G.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 37*x^4 + 143*x^5 + 564*x^6 + 2270*x^7 + 9305*x^8 + 38755*x^9 + 163569*x^10 + 698186*x^11 + 3009129*x^12 + ...
%e A357399 such that A = A(x) satisfies
%e A357399 x = ... + (x*A)^(-2)/(1 - x^(-2))^2 - (x*A)^(-1)/(1 + x^(-1)) + 1 - (x*A)*(1 + x) + (x*A)^2*(1 - x^2)^2 - (x*A)^3*(1 + x^3)^3 + (x*A)^4*(1 - x^4)^4 + ... + (-x*A)^n*(1 - (-x)^n)^n + ...
%e A357399 SPECIFIC VALUES.
%e A357399 A(x) = 2 at x = 0.205304925829036018515173714762264122490105588782437881...
%e A357399 A(1/5) = 1.772537703863275044370749593488314266794544517965356320...
%o A357399 (PARI) {a(n) = my(A=[1]); for(i=1,n, A = concat(A,0);
%o A357399 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 A357399 for(n=0,30,print1(a(n),", "))
%Y A357399 Cf. A357791.
%K A357399 nonn
%O A357399 0,3
%A A357399 _Paul D. Hanna_, Nov 07 2022