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A355353 G.f. A(x) satisfies: 3*x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.

%I A355353 #5 Jun 30 2022 10:39:56
%S A355353 1,3,18,108,660,4275,29106,205377,1485279,10943424,81866493,620316297,
%T A355353 4751289063,36727782675,286153810542,2244799306134,17715992048886,
%U A355353 140560480602810,1120518766292436,8970573523101477,72091628161825608,581375787259765554,4703286596619094686
%N A355353 G.f. A(x) satisfies: 3*x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.
%C A355353 a(n) = Sum_{k=0..n} A355350(n,k) * 3^k for n >= 0.
%H A355353 Paul D. Hanna, <a href="/A355353/b355353.txt">Table of n, a(n) for n = 0..400</a>
%F A355353 G.f. A(x) satisfies:
%F A355353 (1) 3*x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x)^n.
%F A355353 (2) 3*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.
%e A355353 G.f.: A(x) = 1 + 3*x + 18*x^2 + 108*x^3 + 660*x^4 + 4275*x^5 + 29106*x^6 + 205377*x^7 + 1485279*x^8 + 10943424*x^9 + 81866493*x^10 + ...
%e A355353 where
%e A355353 3*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 -+ ...
%e A355353 also,
%e A355353 3*x*P(x) = (1 - x*A(x))*(1 - 1/A(x)) * (1 - x^2*A(x))*(1 - x/A(x)) * (1 - x^3*A(x))*(1 - x^2/A(x)) * (1 - x^4*A(x))*(1 - x^3/A(x)) * ...
%e A355353 where P(x) is the partition function and begins
%e A355353 P(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 + 22*x^8 + 30*x^9 + 42*x^10 + 56*x^11 + 77*x^12 + ... + A000041(n)*x^n + ...
%o A355353 (PARI) {a(n) = my(A=[1,3],t); for(i=1,n, A=concat(A,0); t = ceil(sqrt(2*(#A)+9));
%o A355353 A[#A] = -polcoeff( sum(m=-t,t, (-1)^m*x^(m*(m+1)/2)*Ser(A)^m ), #A-1));A[n+1]}
%o A355353 for(n=0,30,print1(a(n),", "))
%Y A355353 Cf. A355350, A355351, A355352, A355354, A355355, A355356, A355357.
%K A355353 nonn
%O A355353 0,2
%A A355353 _Paul D. Hanna_, Jun 29 2022