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A355363 G.f. A(x) satisfies: 3*x*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.

%I A355363 #8 Jul 03 2025 12:53:19
%S A355363 1,3,27,270,2928,33912,411345,5159337,66364326,870637086,11604385575,
%T A355363 156697653654,2139109221960,29472597414681,409312118499336,
%U A355363 5723853297702444,80528723782556475,1139033786793330429,16187921479930951917,231046413762053945958
%N A355363 G.f. A(x) satisfies: 3*x*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.
%H A355363 Paul D. Hanna, <a href="/A355363/b355363.txt">Table of n, a(n) for n = 0..400</a>
%F A355363 a(n) = Sum_{k=0..n} A355360(n,k) * 3^k for n >= 0.
%F A355363 G.f. A(x) satisfies:
%F A355363 (1) 3*x*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.
%F A355363 (2) -3*x*A(x)^2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) / A(x)^n.
%F A355363 (3) 3*x*A(x)*P(x) = Product_{n>=1} (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)), where P(x) = Product_{n>=1} 1/(1 - x^n) is the partition function (A000041), due to the Jacobi triple product identity.
%F A355363 a(n) ~ c * d^n / n^(3/2), where d = 15.42894386025000237511183711088501557092135179... and c = 0.53592940996364915517082259731565361731654... - _Vaclav Kotesovec_, Jul 03 2025
%e A355363 G.f.: A(x) = 1 + 3*x + 27*x^2 + 270*x^3 + 2928*x^4 + 33912*x^5 + 411345*x^6 + 5159337*x^7 + 66364326*x^8 + 870637086*x^9 + 11604385575*x^10 + ...
%e A355363 where
%e A355363 3*x*A(x) = ... - x^10/A(x)^5 + x^6/A(x)^4 - x^3/A(x)^3 + x/A(x)^2 - 1/A(x) + 1 - x*A(x) + x^3*A(x)^2 - x^6*A(x)^3 + x^10*A(x)^4 -+ ... + (-1)^n * x^(n*(n+1)/2) * A(x)^n + ...
%o A355363 (PARI) {a(n) = my(A=[1,3],t); for(i=1,n, A=concat(A,0); t = ceil(sqrt(2*n+9));
%o A355363 A[#A] = polcoeff( 3*x*Ser(A) - sum(m=-t,t, (-1)^m*x^(m*(m+1)/2)*Ser(A)^m ), #A-1));A[n+1]}
%o A355363 for(n=0,30,print1(a(n),", "))
%Y A355363 Cf. A355360, A355361, A355362, A355364, A355365.
%K A355363 nonn
%O A355363 0,2
%A A355363 _Paul D. Hanna_, Jul 19 2022