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A354645
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G.f. A(x) satisfies: -x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)/2) * A(x)^n.
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3
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1, 1, 3, 9, 23, 62, 179, 571, 1888, 6309, 21114, 71387, 245162, 854703, 3010602, 10673814, 38036229, 136250135, 490696565, 1776089734, 6456234450, 23554296293, 86210280655, 316481218498, 1165089759958, 4300364442451, 15910717070706, 58996259565000
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OFFSET
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0,3
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LINKS
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FORMULA
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G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) -x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)/2) * A(x)^n.
(2) -x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) / A(x)^n.
(3) x = Sum_{n>=0} (-1)^n * x^(n*(n+1)/2) * (A(x)^(n+1) - 1/A(x)^n).
(4) -x = Product_{n>=1} (1 - x^(n-1)*A(x)) * (1 - x^n/A(x)) * (1 - x^n), by the Jacobi Triple Product identity. - Paul D. Hanna, Jan 22 2024
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EXAMPLE
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G.f.: A(x) = 1 + x + 3*x^2 + 9*x^3 + 23*x^4 + 62*x^5 + 179*x^6 + 571*x^7 + 1888*x^8 + 6309*x^9 + 21114*x^10 + 71387*x^11 + 245162*x^12 + ...
such that
x = (A(x) - 1) - x*(A(x)^2 - 1/A(x)) + x^3*(A(x)^3 - 1/A(x)^2) - x^6*(A(x)^4 - 1/A(x)^3) + x^10*(A(x)^5 - 1/A(x)^4) - x^15*(A(x)^6 - 1/A(x)^5) + x^21*(A(x)^7 - 1/A(x)^6) + ...
Also, we have the infinite product
-x = (1 - A(x))*(1 - x/A(x))*(1-x) * (1 - x*A(x))*(1 - x^2/A(x))*(1-x^2) * (1 - x^2*A(x))*(1 - x^3/A(x))*(1-x^3) * (1 - x^3*A(x))*(1 - x^4/A(x))*(1-x^4) * ...
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PROG
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(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff(x + sum(m=-#A, #A, (-1)^m * x^(m*(m-1)/2) * Ser(A)^m ), #A-1)); H=A; A[n+1]}
for(n=0, 40, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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