cp's OEIS Frontend

A356783 Coefficients in the power series A(x) such that: 1 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1) * A(x)^n.

%I A356783 #24 Mar 22 2025 09:37:46
%S A356783 1,1,2,6,17,50,163,525,1770,6066,21154,74787,267371,965233,3513029,
%T A356783 12877687,47499333,176167086,656568385,2457710598,9236079055,
%U A356783 34832753818,131792634266,500121476517,1902979982421,7258942377746,27752992782498,106333425162358,408213503595652
%N A356783 Coefficients in the power series A(x) such that: 1 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1) * A(x)^n.
%C A356783 Related identity: 0 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1).
%H A356783 Paul D. Hanna, <a href="/A356783/b356783.txt">Table of n, a(n) for n = 0..400</a>
%F A356783 G.f. A(x) = Sum_{n>=0} a(n) * x^n satisfies the following relations.
%F A356783 (1) 1 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1) * A(x)^n.
%F A356783 (2) x*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ( (1 - x^(n+1))^n * A(x)^n ).
%F A356783 (3) -x*A(x)^2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (1 - x^(n+1)*A(x))^n.
%F A356783 (4) -A(x)^3 = Sum_{n=-oo..+oo} x^(2*n+1) * (A(x) - x^n)^(n+1) / A(x)^n.
%F A356783 (5) 0 = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n*A(x))^(n+1) / A(x)^n.
%F A356783 (6) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (A(x) - x^(n+1))^n.
%F A356783 a(n) ~ c * d^n / n^(3/2), where d = 4.04962821886295599791727073173857... and c = 0.613483546803830745310382482744... - _Vaclav Kotesovec_, Mar 22 2025
%e A356783 G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 17*x^4 + 50*x^5 + 163*x^6 + 525*x^7 + 1770*x^8 + 6066*x^9 + 21154*x^10 + 74787*x^11 + 267371*x^12 + ...
%e A356783 such that
%e A356783 1 = ... + x^(-3)*(1 - x^(-2))^(-1)/A(x)^2 + x^(-1)/A(x) + x*0 + x^3*(1 - x)^2*A(x) + x^5*(1 - x^2)^3*A(x)^2 + x^7*(1 - x^3)^4*A(x)^3 + ... + x^(2*n+1)*(1 - x^n)^(n+1)*A(x)^n + ...
%e A356783 also
%e A356783 -A(x)^3 = ... + x^(-3)*(A(x) - x^(-2))^(-1)*A(x)^2 + x^(-1)*A(x) + x*(A(x) - 1) + x^3*(A(x) - x)^2/A(x) + x^5*(1 - x^2)^3/A(x)^2 + x^7*(A(x) - x^3)^4/A(x)^3 + ... + x^(2*n+1)*(A(x) - x^n)^(n+1)/A(x)^n + ...
%o A356783 (PARI) {a(n) = my(A=[1]); for(i=0,n, A = concat(A,0);
%o A356783 A[#A] = polcoeff(1 - sum(n=-#A\2-1,#A\2+1, x^(2*n+1) * (1 - x^n +x*O(x^#A))^(n+1) * Ser(A)^n  ),#A-2); );A[n+1]}
%o A356783 for(n=0,30, print1(a(n),", "))
%Y A356783 Cf. A357151, A357152, A357153, A357154, A357155.
%Y A356783 Cf. A357200, A357400, A357402, A357403, A357404, A357405.
%K A356783 nonn
%O A356783 0,3
%A A356783 _Paul D. Hanna_, Sep 15 2022