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A380065 G.f. A(x) satisfies 2 = Sum_{n=-oo..+oo} x^(2*n) * (A(x) - x^(2*n+1))^(n-1).

%I A380065 #12 Jan 25 2025 05:39:13
%S A380065 1,3,14,98,785,6702,59968,554872,5266164,50982561,501511295,
%T A380065 4998413255,50366515829,512257729704,5251739403578,54216242355803,
%U A380065 563112825354739,5880229017352112,61697782946493598,650137346153943901,6877281106762452016,73003969200802059386,777423197278368997747
%N A380065 G.f. A(x) satisfies 2 = Sum_{n=-oo..+oo} x^(2*n) * (A(x) - x^(2*n+1))^(n-1).
%H A380065 Paul D. Hanna, <a href="/A380065/b380065.txt">Table of n, a(n) for n = 0..300</a>
%F A380065 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F A380065 (1) 2 = Sum_{n=-oo..+oo} x^(2*n) * (A(x) - x^(2*n+1))^(n-1).
%F A380065 (2) 2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(2*n+3)) / (1 - x^(2*n+1)*A(x))^(n+2).
%F A380065 a(n) ~ c * d^n / n^(3/2), where d = 11.3925137162742418189953728259701493179421085245... and c = 0.47865899177399443140172857981961910025627907... - _Vaclav Kotesovec_, Jan 25 2025
%F A380065 A(r) = 1.9022130897341229314214706611442588... where r = 0.087776940621234174350691113899657441078036058... (r = 1/d using d given above). - _Paul D. Hanna_, Jan 25 2025
%e A380065 G.f.: A(x) = 1 + 3*x + 14*x^2 + 98*x^3 + 785*x^4 + 6702*x^5 + 59968*x^6 + 554872*x^7 + 5266164*x^8 + 50982561*x^9 + 501511295*x^10 + ...
%e A380065 SPECIFIC VALUES.
%e A380065 A(t) = 11/6 at t = 0.087620332691056981136009348051386438683729069234188...
%e A380065 A(t) = 9/5 at t = 0.0874182043261646741149981956172218350826689730404996...
%e A380065 A(t) = 7/4 at t = 0.0869318434801806744979925380398173982849232718054976...
%e A380065 A(t) = 5/3 at t = 0.0855310829705482164598754145517095235059157760682890...
%e A380065 A(t) = 3/2 at t = 0.0796031658964171493942046069785384177317948916037014...
%e A380065   where 2 = Sum_{n=-oo..+oo} t^(2*n) * (3/2 - t^(2*n+1))^(n-1).
%e A380065 A(t) = 4/3 at t = 0.0669971124636040282524670754433309879051494329729690...
%e A380065 A(t) = 5/4 at t = 0.0567962686348021793992058668495458301112625095350602...
%e A380065 A(t) = 6/5 at t = 0.0489819942637972068395484924808191716360116583798063...
%e A380065   where 2 = Sum_{n=-oo..+oo} t^(2*n) * (6/5 - t^(2*n+1))^(n-1).
%e A380065 A(1/12) = 1.58752189609489937448603644291125612645131622024866...
%e A380065   where 2 = Sum_{n=-oo..+oo} (1/12)^(2*n) * (A(1/12) - 1/12^(2*n+1))^(n-1).
%e A380065 A(1/13) = 1.45355309884363897926210438733196707095664015067023...
%e A380065 A(1/14) = 1.37996515081773437231256208297748336847364447973019...
%e A380065 A(1/15) = 1.33017535921565190525655216189827503321523887201567...
%e A380065 A(1/16) = 1.29333445310054157832439323915232565378414742432913...
%e A380065 A(1/20) = 1.20602500277090859861285926459871218765300896393649...
%e A380065   where 2 = Sum_{n=-oo..+oo} (1/20)^(2*n) * (A(1/20) - 1/20^(2*n+1))^(n-1).
%o A380065 (PARI) {a(n) = my(V=[1]); for(i=1, n, V=concat(V, 0); A = Ser(V);
%o A380065 V[#V] = polcoef(-2 + sum(n=-#V, #V, x^(2*n) * (A - x^(2*n+1))^(n-1) ), #V-1) ); V[n+1]}
%o A380065 for(n=0, 30, print1(a(n), ", "))
%Y A380065 Cf. A379765.
%K A380065 nonn
%O A380065 0,2
%A A380065 _Paul D. Hanna_, Jan 24 2025