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A384824 G.f. A(x) satisfies 1/x^19 = Sum_{n=-oo..+oo} A(x)^n * x^n * (1 - x^n)^(n+8).

%I A384824 #10 Jun 11 2025 03:10:05
%S A384824 1,1,5,38,319,2871,27507,273925,2808973,29457644,314470771,3405995019,
%T A384824 37334767867,413397265017,4617060957512,51951448775027,
%U A384824 588371324004508,6701761863368579,76723673176823126,882342098781937683,10188542630975395255,118082022786322630334,1373108879790849494070
%N A384824 G.f. A(x) satisfies 1/x^19 = Sum_{n=-oo..+oo} A(x)^n * x^n * (1 - x^n)^(n+8).
%H A384824 Paul D. Hanna, <a href="/A384824/b384824.txt">Table of n, a(n) for n = 0..300</a>
%F A384824 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F A384824 (1) 1/x^19 = Sum_{n=-oo..+oo} A(x)^n * x^n * (1 - x^n)^(n+8).
%F A384824 (2) x = Sum_{n=-oo..+oo, n<>0} (-1/A(x))^n * x^((n-4)*(n-5)) / (1 - x^n)^(n-8).
%F A384824 a(n) ~ c * d^n / n^(3/2), where d = 12.46033620173328231233579215988893957838459959... and c = 0.113752375605091798753361983956448030623... - _Vaclav Kotesovec_, Jun 11 2025
%e A384824 G.f.: A(x) = 1 + x + 5*x^2 + 38*x^3 + 319*x^4 + 2871*x^5 + 27507*x^6 + 273925*x^7 + 2808973*x^8 + 29457644*x^9 + 314470771*x^10 + ...
%o A384824 (PARI) {a(n) = my(A=[1,1,0,0,0,0]); for(i=1, n, A = concat(A, 0);
%o A384824 A[#A-4] = -polcoeff( sum(m=-#A, #A, x^m * Ser(A)^m * (1 - x^m +x*O(x^n))^(m+8) ), #A-25)); A[n+1]}
%o A384824 for(n=0, 30, print1(a(n), ", "))
%Y A384824 Cf. A366731, A384821, A384822, A384823, A384825, A384826, A384827, A384828.
%K A384824 nonn
%O A384824 0,3
%A A384824 _Paul D. Hanna_, Jun 10 2025