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 A380712 #12 Mar 26 2025 12:14:53
%S A380712 1,8,84,1040,14220,207416,3163352,49838112,804826128,13251624272,
%T A380712 221630530572,3754763811696,64301286803888,1111314020855608,
%U A380712 19358763742909840,339542985410593024,5991328544544083368,106282296849129147080,1894330721630908390908,33907409814314990430864
%N A380712 G.f. A(x) satisfies 1/2 = Sum_{n=-oo..+oo} (-1)^(n-1) * x^(2*n) * (A(x) + x^n)^(n-1).
%H A380712 Paul D. Hanna, <a href="/A380712/b380712.txt">Table of n, a(n) for n = 0..300</a>
%F A380712 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F A380712 (1) 1/2 = Sum_{n=-oo..+oo} (-1)^(n-1) * x^(2*n) * (A(x) + x^n)^(n-1).
%F A380712 (2) 1/2 = Sum_{n=-oo..+oo} (-1)^(n-1) * x^(n*(n-1)) / (1 + x^n*A(x))^(n+1).
%e A380712 G.f.: A(x) = 1 + 8*x + 84*x^2 + 1040*x^3 + 14220*x^4 + 207416*x^5 + 3163352*x^6 + 49838112*x^7 + 804826128*x^8 + 13251624272*x^9 + ...
%e A380712 where 1/2 = Sum_{n=-oo..+oo} (-1)^(n-1) * x^(2*n) * (A(x) + x^n)^(n-1).
%e A380712 SPECIFIC VALUES.
%e A380712 A(t) = 11/4 at t = 0.0516760605367732994895781933476882835083123366991550...
%e A380712 A(t) = 5/2 at t = 0.0512706205731010752236248787488564353175797944139758...
%e A380712 A(t) = 9/4 at t = 0.0500962358404660270765237205868976107655318520352114...
%e A380712 A(t) = 2 at t = 0.0477293025632112771125356682128360981725295094219849155...
%e A380712 A(1/20) = 2.23597314331862522198845196504910911754398888561681...
%e A380712 A(1/21) = 1.99138007575616089662630287849407074435849433028712...
%e A380712 A(1/22) = 1.84963311768563536747915607573430165080208182054100...
%e A380712 A(1/23) = 1.75061850078240549448668878592242454463377302943692...
%e A380712 A(1/24) = 1.67559723922061838865975191745342891467177127438988...
%o A380712 (PARI) {a(n) = my(V=[1]); for(i=1, n, V=concat(V, 0); A = Ser(V);
%o A380712 V[#V] = polcoef(2 + 4*sum(n=-#V, #V, (-1)^n * x^(2*n) * (A + x^n)^(n-1) ), #V-1) ); V[n+1]}
%o A380712 for(n=0, 30, print1(a(n), ", "))
%Y A380712 Cf. A379763, A380068.
%K A380712 nonn
%O A380712 0,2
%A A380712 _Paul D. Hanna_, Feb 21 2025