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A370341
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Expansion of g.f. A(x) satisfying Sum_{n>=0} (x^n + A(x))^n = Product_{n>=1} (1 - x^(2*n)) * (1 + x^n + A(x))^2 / (1 + x^(2*n) + A(x))^2.
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4
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1, -3, 9, -35, 158, -784, 4120, -22477, 125884, -719090, 4172414, -24521300, 145659770, -873125958, 5274788402, -32083526351, 196312270733, -1207543838993, 7462752787850, -46315370172411, 288536968531982, -1803738499481171, 11311175379539531, -71135663269605615, 448549432404511902
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OFFSET
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1,2
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COMMENTS
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Compare to identity: 1 + 2*Sum_{n>=0} x^(n^2) = Product_{n>=1} (1 - x^(2*n)) * (1 + x^n)^2 / (1 + x^(2*n))^2.
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LINKS
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FORMULA
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G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) B(x) = Product_{n>=1} (1 - x^(2*n)) * (1 + x^n + A(x))^2 / (1 + x^(2*n) + A(x))^2.
(2) B(x) = Sum_{n>=0} (x^n + A(x))^n.
(3) B(x) = Sum_{n>=0} x^(n^2) / (1 - x^n*A(x))^(n+1).
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EXAMPLE
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G.f.: A(x) = x - 3*x^2 + 9*x^3 - 35*x^4 + 158*x^5 - 784*x^6 + 4120*x^7 - 22477*x^8 + 125884*x^9 - 719090*x^10 + 4172414*x^11 - 24521300*x^12 + ...
By definition, A = A(x) allows for the following expressions to equal
B(x) = 1 + (x + A) + (x^2 + A)^2 + (x^3 + A)^3 + (x^4 + A)^4 + (x^5 + A)^5 + ...
B(x) = (1 - x^2)*(1 + x + A)^2/(1 + x^2 + A)^2 * (1 - x^4)*(1 + x^2 + A)^2/(1 + x^4 + A)^2 * (1 - x^6)*(1 + x^3 + A)^2/(1 + x^6 + A)^2 * (1 - x^8)*(1 + x^4 + A)^2/(1 + x^8 + A)^2 * ...
where B(x) begins
B(x) = 1 + 2*x - 2*x^2 + 6*x^3 - 21*x^4 + 98*x^5 - 483*x^6 + 2514*x^7 - 13594*x^8 + 75624*x^9 - 429833*x^10 + 2484632*x^11 - 14559106*x^12 + ...
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PROG
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(PARI) {a(n) = my(A=[0, 1]); for(i=1, n, A=concat(A, 0);
A[#A] = -polcoeff( sum(m=0, #A, (x^m + Ser(A))^m ) - prod(m=1, #A, (1 - x^(2*m))*(1 + x^m + Ser(A))^2/(1 + x^(2*m) + Ser(A))^2 ), #A-1)); A[n+1]}
for(n=1, 30, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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