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A384821 G.f. A(x) satisfies -1/x = Sum_{n=-oo..+oo} A(x)^n * x^n * (1 - x^n)^(n+2).

%I A384821 #13 Jun 11 2025 02:59:32
%S A384821 1,2,5,22,91,416,1978,9738,49181,253572,1328528,7053672,37866294,
%T A384821 205188765,1120824743,6165155890,34119043994,189839648588,
%U A384821 1061344406923,5959197795092,33588952625106,189986944364176,1078034452020854,6134848540680166,35005230073846833,200229444332667654
%N A384821 G.f. A(x) satisfies -1/x = Sum_{n=-oo..+oo} A(x)^n * x^n * (1 - x^n)^(n+2).
%H A384821 Paul D. Hanna, <a href="/A384821/b384821.txt">Table of n, a(n) for n = 0..401</a>
%F A384821 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F A384821 (1) -1/x = Sum_{n=-oo..+oo} A(x)^n * x^n * (1 - x^n)^(n+2).
%F A384821 (2) -x = Sum_{n=-oo..+oo, n<>0} (-1/A(x))^n * x^((n-1)*(n-2)) / (1 - x^n)^(n-2).
%F A384821 a(n) ~ c * d^n / n^(3/2), where d = 6.07021478936467894926862346663483720359... and c = 0.6881950589132830412100382237325446... - _Vaclav Kotesovec_, Jun 11 2025
%e A384821 G.f.: A(x) = 1 + 2*x + 5*x^2 + 22*x^3 + 91*x^4 + 416*x^5 + 1978*x^6 + 9738*x^7 + 49181*x^8 + 253572*x^9 + 1328528*x^10 + ...
%e A384821 SPECIFIC VALUES.
%e A384821 A(t) = 2 at t = 0.162924020448782314256916956456618618555937137963260...
%e A384821 A(t) = 9/5 at t = 0.15713093477961462528780113190237390843002535981643...
%e A384821 A(t) = 8/5 at t = 0.14467881602482935797425598908263109752382579929421...
%e A384821 A(t) = 3/2 at t = 0.13461615563760120581581313629107981605312435881819...
%e A384821 A(t) = 4/3 at t = 0.10915621052082212882653574706851509193398803739915...
%e A384821 A(1/7) = 1.5793911503434252677981671019480264164820055324466...
%e A384821 A(1/8) = 1.4268350851974567615394958810072981944850896947894...
%e A384821 A(1/9) = 1.3435470274993477728207146854713823085043981519155...
%e A384821 A(1/10) = 1.2892440747830023480637465318368592024118039394009...
%e A384821 A(1/11) = 1.2505209808081799972669805855553805055082827658365...
%o A384821 (PARI) {a(n) = my(A=[1,2,0]); for(i=1, n, A = concat(A, 0);
%o A384821 A[#A-1] = polcoeff( sum(m=-#A, #A, x^m * Ser(A)^m * (1 - x^m +x*O(x^n))^(m+2) ), #A-4)); A[n+1]}
%o A384821 for(n=0, 30, print1(a(n), ", "))
%Y A384821 Cf. A366731, A384822, A384823, A384824, A384825, A384826, A384827, A384828.
%K A384821 nonn
%O A384821 0,2
%A A384821 _Paul D. Hanna_, Jun 10 2025