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A366731 Expansion of g.f. A(x) satisfying 0 = Sum_{n=-oo..+oo} x^n * A(x)^n * (1 - x^(n-1))^(n+1).

%I A366731 #9 Jun 11 2025 02:43:23
%S A366731 1,1,0,2,2,6,19,41,99,307,750,2062,5776,15674,43700,123729,345728,
%T A366731 982580,2801615,7994268,22953104,66128105,190846074,552959720,
%U A366731 1605817449,4673526011,13635237816,39860703465,116739997283,342538898105,1006709394181,2963267980415,8735388348630
%N A366731 Expansion of g.f. A(x) satisfying 0 = Sum_{n=-oo..+oo} x^n * A(x)^n * (1 - x^(n-1))^(n+1).
%C A366731 a(n) = Sum_{k=0..n} A366730(n,k) for n >= 0.
%H A366731 Paul D. Hanna, <a href="/A366731/b366731.txt">Table of n, a(n) for n = 0..300</a>
%F A366731 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F A366731 (1) 0 = Sum_{n=-oo..+oo} x^n * A(x)^n * (1 - x^(n-1))^(n+1).
%F A366731 (2) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ( A(x)^n * (1 - x^(n+1))^(n-1) ).
%F A366731 a(n) ~ c * d^n / n^(3/2), where d = 3.087019811495... and c = 0.3580397646... - _Vaclav Kotesovec_, Jun 11 2025
%e A366731 G.f.: A(x) = 1 + x + 2*x^3 + 2*x^4 + 6*x^5 + 19*x^6 + 41*x^7 + 99*x^8 + 307*x^9 + 750*x^10 + 2062*x^11 + 5776*x^12 + 15674*x^13 + 43700*x^14 + ...
%o A366731 (PARI) {a(n) = my(A=[1]); for(i=1,n, A = concat(A,0);
%o A366731 A[#A] = polcoeff( sum(n=-#A,#A, x^n * Ser(A)^n * (1 - x^(n-1))^(n+1) ), #A-2));A[n+1]}
%o A366731 for(n=0,40,print1(a(n),", "))
%Y A366731 Cf. A366730, A366732, A366733, A366734, A366735.
%K A366731 nonn
%O A366731 0,4
%A A366731 _Paul D. Hanna_, Oct 29 2023