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A355362 G.f. A(x) satisfies: 2*x*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.

%I A355362 #9 Jul 03 2025 12:50:37
%S A355362 1,2,14,106,852,7286,65216,603714,5731930,55506348,546091942,
%T A355362 5443033448,54845812094,557774491672,5717718435034,59017814463718,
%U A355362 612873311614338,6398538141213916,67121038262747380,707114126290890810,7478082640450505012,79360375914717108922
%N A355362 G.f. A(x) satisfies: 2*x*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.
%H A355362 Paul D. Hanna, <a href="/A355362/b355362.txt">Table of n, a(n) for n = 0..400</a>
%F A355362 a(n) = Sum_{k=0..n} A355360(n,k) * 2^k for n >= 0.
%F A355362 G.f. A(x) satisfies:
%F A355362 (1) 2*x*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.
%F A355362 (2) -2*x*A(x)^2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) / A(x)^n.
%F A355362 (3) 2*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 A355362 a(n) ~ c * d^n / n^(3/2), where d = 11.38813717738101179115221618346020026348459... and c = 0.5257715220992591718905720654742321646... - _Vaclav Kotesovec_, Jul 03 2025
%e A355362 G.f.: A(x) = 1 + 2*x + 14*x^2 + 106*x^3 + 852*x^4 + 7286*x^5 + 65216*x^6 + 603714*x^7 + 5731930*x^8 + 55506348*x^9 + 546091942*x^10 + ...
%e A355362 where
%e A355362 2*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 A355362 (PARI) {a(n) = my(A=[1,2],t); for(i=1,n, A=concat(A,0); t = ceil(sqrt(2*n+9));
%o A355362 A[#A] = polcoeff( 2*x*Ser(A) - sum(m=-t,t, (-1)^m*x^(m*(m+1)/2)*Ser(A)^m ), #A-1));A[n+1]}
%o A355362 for(n=0,30,print1(a(n),", "))
%Y A355362 Cf. A355360, A355361, A355363, A355364, A355365.
%K A355362 nonn
%O A355362 0,2
%A A355362 _Paul D. Hanna_, Jul 19 2022