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A384822 G.f. A(x) satisfies 1/x^5 = Sum_{n=-oo..+oo} A(x)^n * x^n * (1 - x^n)^(n+4).

%I A384822 #10 Jun 11 2025 03:05:09
%S A384822 1,1,5,19,109,598,3592,22110,140467,911136,6014277,40260501,272682397,
%T A384822 1865181921,12866239311,89403333632,625211046931,4396844409898,
%U A384822 31075863324446,220618909826500,1572549447431889,11249693613964519,80743512234554655,581272589032594530,4196118995069449989
%N A384822 G.f. A(x) satisfies 1/x^5 = Sum_{n=-oo..+oo} A(x)^n * x^n * (1 - x^n)^(n+4).
%H A384822 Paul D. Hanna, <a href="/A384822/b384822.txt">Table of n, a(n) for n = 0..300</a>
%F A384822 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F A384822 (1) 1/x^5 = Sum_{n=-oo..+oo} A(x)^n * x^n * (1 - x^n)^(n+4).
%F A384822 (2) x = Sum_{n=-oo..+oo, n<>0} (-1/A(x))^n * x^((n-2)*(n-3)) / (1 - x^n)^(n-4).
%F A384822 a(n) ~ c * d^n / n^(3/2), where d = 7.687452504111926947946743863677977... and c = 0.2779337748307189293421456249838... - _Vaclav Kotesovec_, Jun 11 2025
%e A384822 G.f.: A(x) = 1 + x + 5*x^2 + 19*x^3 + 109*x^4 + 598*x^5 + 3592*x^6 + 22110*x^7 + 140467*x^8 + 911136*x^9 + 6014277*x^10 + ...
%o A384822 (PARI) {a(n) = my(A=[1,1,0,0]); for(i=1, n, A = concat(A, 0);
%o A384822 A[#A-2] = -polcoeff( sum(m=-#A, #A, x^m * Ser(A)^m * (1 - x^m +x*O(x^n))^(m+4) ), #A-9)); A[n+1]}
%o A384822 for(n=0, 30, print1(a(n), ", "))
%Y A384822 Cf. A366731, A384821, A384823, A384824, A384825, A384826, A384827, A384828.
%K A384822 nonn
%O A384822 0,3
%A A384822 _Paul D. Hanna_, Jun 10 2025