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G.f. A(x) satisfies: 2 = Sum_{n=-oo..+oo} x^(n*(n+1)/2) * A(x)^n.
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%I #8 Aug 21 2023 11:12:00

%S 1,2,2,12,22,144,318,2102,5120,34274,88352,597002,1599676,10879502,

%T 29983958,204851678,576914820,3953960052,11329537402,77815428652,

%U 226170428918,1555598157856,4576144621100,31500863667990,93634976287220,644808182456240,1934219875423410

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

%H Paul D. Hanna, <a href="/A355871/b355871.txt">Table of n, a(n) for n = 0..500</a>

%F G.f. A(x,y) = Sum_{n>=0} a(n)*x^n satisfies:

%F (1) 2 = Sum_{n=-oo..+oo} x^(n*(n+1)/2) * A(x)^n.

%F (2) 2 = Product_{n>=1} (1 + x^n*A(x)) * (1 + x^(n-1)/A(x)) * (1 - x^n), by the Jacobi triple product identity.

%F (3) a(n) = Sum_{k=0..3*n} A355870(n,k)*2^k for n >= 0.

%e G.f.: A(x) = 1 + 2*x + 2*x^2 + 12*x^3 + 22*x^4 + 144*x^5 + 318*x^6 + 2102*x^7 + 5120*x^8 + 34274*x^9 + 88352*x^10 + 597002*x^11 + 1599676*x^12 + ...

%e where

%e 2 = ... + 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 + ... + x^(n*(n+1)/2) * A(x)^n + ...

%e Also,

%e 2 = (1 + x*A(x))*(1 + 1/A(x))*(1-x) * (1 + x^2*A(x))*(1 + x/A(x))*(1-x^2) * (1 + x^3*A(x))*(1 + x^2/A(x))*(1-x^3) * (1 + x^4*A(x))*(1 + x^3/A(x))*(1-x^4) * ...

%e Specific values.

%e A(1/5) = 1.8349253975...

%o (PARI) {a(n) = my(A=[1],M); for(i=1,n, A=concat(A,0); M = ceil(sqrt(2*(#A)+9));

%o A[#A] = polcoeff( -2 + sum(m=-M,M, x^(m*(m+1)/2) * Ser(A)^m ), #A-1));A[n+1]}

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A355870.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jul 19 2022