This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A380557 #8 Feb 03 2025 13:51:21
%S A380557 1,1,2,10,35,146,589,2521,10880,48130,215490,978131,4483493,20740309,
%T A380557 96667511,453596099,2140879339,10157274086,48414142443,231726319442,
%U A380557 1113290775079,5366873616498,25952658569610,125856499026093,611930422986515,2982444057333882,14568259180879990,71307949455547118
%N A380557 G.f. satisfies A(x) such that: -1 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n+1) * (1 + x^n)^(n+1) * A(x)^n.
%C A380557 Related identity: 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n+1) * (1 + x^n)^(n+1).
%C A380557 Note that formulas (5) and (6) are identities, and hold true for all A(x).
%H A380557 Paul D. Hanna, <a href="/A380557/b380557.txt">Table of n, a(n) for n = 0..400</a>
%F A380557 G.f. A(x) = Sum_{n>=0} a(n) * x^n satisfies the following relations.
%F A380557 (1) -1 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n+1) * (1 + x^n)^(n+1) * A(x)^n.
%F A380557 (2) A(x)^3 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n+1) * (A(x) + x^n)^(n+1) / A(x)^n.
%F A380557 (3) -1 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-3)+1) / ( (1 + x^n)^(n-1) * A(x)^n ).
%F A380557 (4) A(x)^3 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-3)+1) * A(x)^n / (1 + x^n*A(x))^(n-1).
%F A380557 (5) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n+1) * (1 + x^n*A(x))^(n+1) / A(x)^n.
%F A380557 (6) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-3)+1) * A(x)^n / (A(x) + x^n)^(n-1).
%e A380557 G.f.: A(x) = 1 + x + 2*x^2 + 10*x^3 + 35*x^4 + 146*x^5 + 589*x^6 + 2521*x^7 + 10880*x^8 + 48130*x^9 + 215490*x^10 + ...
%e A380557 SPECIFIC VALUES.
%e A380557 A(t) = 15/8 at t = 0.19344501240894726710748422307503613491843235983978...
%e A380557 where -1 = Sum_{n=-oo..+oo} (-1)^n * t^(2*n+1) * (1 + t^n)^(n+1) * (15/8)^n.
%e A380557 A(t) = 13/7 at t = 0.19341948378562934846535490742010025888491204467175...
%e A380557 A(t) = 11/6 at t = 0.19333502076470314454576717568898264806286912451280...
%e A380557 A(t) = 9/5 at t = 0.193110973645115287451084966528093291445869685605026...
%e A380557 A(t) = 7/4 at t = 0.192511645242345015361112270688385360547743653185979...
%e A380557 A(t) = 5/3 at t = 0.190649553303712199475798636706794101316873079578727...
%e A380557 A(t) = 3/2 at t = 0.182089586086018008207410926078691444238166561231377...
%e A380557 A(t) = 4/3 at t = 0.161675866655112310035152981730415472582224089685922...
%e A380557 A(t) = 5/4 at t = 0.143001255997678107192529149738503806026990657450325...
%e A380557 A(t) = 6/5 at t = 0.127286533527611786785145642678412658294861536180040...
%e A380557 A(t) = 7/6 at t = 0.114247661034580905508079370420172649525885310579285...
%e A380557 A(1/6) = 1.3631240552377275579566206545056633589020532732074...
%e A380557 where -1 = Sum_{n=-oo..+oo} (-1)^n * (1/6)^(2*n+1) * (1 + (1/6)^n)^(n+1) * A(1/6)^n.
%e A380557 A(1/7) = 1.2494768685846922246570903862376666502561254090745...
%e A380557 A(1/8) = 1.1937090558071312140144246419584402049019509862828...
%e A380557 A(1/9) = 1.1594652229281839152092617957390758203214375656645...
%e A380557 A(1/10) = 1.135997746902180909378823046338236460433675615420...
%o A380557 (PARI) {a(n) = my(A=[1]); for(i=0, n, A = concat(A, 0);
%o A380557 A[#A] = polcoeff(1 + sum(m=-#A, #A, (-1)^m * x^(2*m+1) * (1 + x^m +x*O(x^#A))^(m+1) * Ser(A)^m ), #A-2); ); A[n+1]}
%o A380557 for(n=0, 30, print1(a(n), ", "))
%Y A380557 Cf. A356783.
%K A380557 nonn
%O A380557 0,3
%A A380557 _Paul D. Hanna_, Feb 03 2025