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A378264 G.f. A(x) satisfies 1/x = Sum_{n=-oo..+oo} A(x)^n * (1 + A(x)^n)^(n+1).

%I A378264 #12 Dec 20 2024 23:38:00
%S A378264 1,3,10,38,164,783,4005,21400,117602,659019,3748736,21588796,
%T A378264 125646501,737977155,4369147468,26048215099,156249597852,942344615209,
%U A378264 5710710976884,34756875588376,212361179832431,1302068876523950,8009024360554817,49407447276951470,305609996146288873,1895015255546957578
%N A378264 G.f. A(x) satisfies 1/x = Sum_{n=-oo..+oo} A(x)^n * (1 + A(x)^n)^(n+1).
%C A378264 Related identity: Sum_{n=-oo..+oo} x^n*(y - x^n)^n = 0, which holds formally for all y.
%H A378264 Paul D. Hanna, <a href="/A378264/b378264.txt">Table of n, a(n) for n = 1..366</a>
%F A378264 G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
%F A378264 (1) 1/x = Sum_{n=-oo..+oo} A(x)^n * (1 + A(x)^n)^(n+1).
%F A378264 (2) A(x) = x * Sum_{n=-oo..+oo} A(x)^(n^2) / (1 + A(x)^(n+1))^n.
%F A378264 From _Paul D. Hanna_, Dec 20 2024: (Start)
%F A378264 (3) 1/x = Sum_{n=-oo..+oo} A(x)^(2*n) * (A(x)^n - 1)^n.
%F A378264 (4) A(x) = x * Sum_{n=-oo..+oo, n <> -1} A(x)^(n^2) / (1 - A(x)^(n+1))^(n+1).
%F A378264 (5) A(B(x)) = x where B(x) = 1/( Sum_{n=-oo..+oo} x^n * (x^n + 1)^(n+1) ).
%F A378264 (End)
%e A378264 G.f.: A(x) = x + 3*x^2 + 10*x^3 + 38*x^4 + 164*x^5 + 783*x^6 + 4005*x^7 + 21400*x^8 + 117602*x^9 + 659019*x^10 + 3748736*x^11 + 21588796*x^12 + ...
%e A378264 SPECIFIC VALUES.
%e A378264 A(t) = 1/3 at t = 0.14832728317680424382350400745104642263167027946862...
%e A378264 A(t) = 1/4 at t = 0.13433913917600443178696714330960568436967435856815...
%e A378264 A(t) = 1/5 at t = 0.12029812285398972879219940261295281978412524937754...
%e A378264 A(3/20) = 0.3521325903099608361455770617898033111722103407971...
%e A378264 A(1/7) = 0.29252723487814042698570516039406838227427731852655...
%e A378264 A(1/8) = 0.21500724214149512130643660913381998900575603076452...
%e A378264 A(1/9) = 0.17407688053908806913569913139334508111874650183559...
%e A378264 A(1/10) = 0.14711097488062849474543678333471254427936118296317...
%o A378264 (PARI) {a(n) = my(V=[0,1],A); for(i=1,n, V=concat(V,0); A = Ser(V);
%o A378264 V[#V] = polcoef( sum(m=-#A,#A, A^m*(1 + A^m)^(m+1) ), #V-3); ); polcoef(A,n)}
%o A378264 for(n=1,40,print1(a(n),", "))
%Y A378264 Cf. A379200, A379199, A166952, A379202, A379203, A379204, A379205.
%K A378264 nonn
%O A378264 1,2
%A A378264 _Paul D. Hanna_, Dec 08 2024