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A354652
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G.f. A(x) satisfies: -2 = Sum_{n=-oo..oo} (-1)^n * x^(n*(n+1)/2) * A(x)^(n*(n-1)/2).
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9
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3, 26, 702, 24312, 964654, 41438412, 1876038114, 88154317378, 4258925591364, 210228411365958, 10556622328639744, 537564689914558410, 27693960347082015456, 1440798064785384773930, 75590961232091579641890, 3994794446280096850372038, 212460780898577846286309772
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OFFSET
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0,1
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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) -2 = Sum_{n=-oo..oo} (-1)^n * x^(n*(n-1)/2) * A(x)^(n*(n+1)/2).
(2) -2 = Sum_{n>=0} (-1)^n * x^(n*(n-1)/2) * (1 - x^(2*n+1)) * A(x)^(n*(n+1)/2).
(3) -2 = Sum_{n>=0} (-1)^n * A(x)^(n*(n-1)/2) * (1 - A(x)^(2*n+1)) * x^(n*(n+1)/2).
(4) -2 = Product_{n>=1} (1 - x^n*A(x)^n) * (1 - x^(n-1)*A(x)^n) * (1 - x^n*A(x)^(n-1)), by the Jacobi triple product identity.
a(n) = (-1)^(n+1) * Sum_{k=0..2*n+1} A354649(n,k)*(-2)^k, for n >= 0.
a(n) = Sum_{k=0..2*n+1} A354650(n,k)*2^k, for n >= 0.
a(n) ~ c * d^n / n^(3/2), where d = 58.4550529987436715308576941861333478321842744514285703940141525... and c = 0.757806366482059336893833755732847891553108700077361984219509... - Vaclav Kotesovec, Jun 08 2022
Formula (4) can be rewritten as the functional equation QPochhammer(x*y) * QPochhammer(1/x, x*y)/(1 - 1/x) * QPochhammer(1/y, x*y)/(1 - 1/y) = -2. - Vaclav Kotesovec, Jan 19 2024
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EXAMPLE
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G.f.: A(x) = 3 + 26*x + 702*x^2 + 24312*x^3 + 964654*x^4 + 41438412*x^5 + 1876038114*x^6 + 88154317378*x^7 + 4258925591364*x^8 + ...
such that A = A(x) satisfies:
(1) -2 = ... + x^36*A^28 - x^28*A^21 + x^21*A^15 - x^15*A^10 + x^10*A^6 - x^6*A^3 + x^3*A - x + 1 - A + x*A^3 - x^3*A^6 + x^6*A^10 - x^10*A^15 + x^15*A^21 - x^21*A^28 + x^28*A^36 + ...
(2) -2 = (1-x) - (1-x^3)*A + x*(1-x^5)*A^3 - x^3*(1-x^7)*A^6 + x^6*(1-x^9)*A^10 - x^10*(1-x^11)*A^15 + x^15*(1-x^13)*A^21 - x^21*(1-x^15)*A^28 + ...
(3) -2 = (1-A) - (1-A^3)*x + A*(1-A^5)*x^3 - A^3*(1-A^7)*x^6 + A^6*(1-A^9)*x^10 - A^10*(1-A^11)*x^15 + A^15*(1-A^13)*x^21 - A^21*(1-A^15)*x^28 + ...
(4) -2 = (1 - x*A)*(1 - A)*(1-x) * (1 - x^2*A^2)*(1 - x*A^2)*(1 - x^2*A) * (1 - x^3*A^3)*(1 - x^2*A^3)*(1 - x^3*A^2) * (1 - x^4*A^4)*(1 - x^3*A^4)*(1 - x^4*A^3) * (1 - x^5*A^5)*(1 - x^4*A^5)*(1 - x^5*A^4) * ...
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MATHEMATICA
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(* Calculation of constant d: *) 1/r /. FindRoot[{r*s * QPochhammer[1/r, r*s] * QPochhammer[1/s, r*s] * QPochhammer[r*s] / ((-1 + r)*(-1 + s)) == -2, -2*(-1 + r)*(-1 + s)*Log[r*s] * Derivative[0, 1][QPochhammer][1/r, r*s] / QPochhammer[1/r, r*s] + r*s*Log[r*s] * QPochhammer[1/r, r*s] * QPochhammer[r*s, r*s] * Derivative[0, 1][QPochhammer][1/s, r*s] + (2*(-1 + r)*(QPochhammer[r*s, r*s]*(Log[r*s] + (-1 + s)*QPolyGamma[0, 1, r*s] - (-1 + s)* QPolyGamma[0, -Log[s]/Log[r*s], r*s]) - r*(-1 + s)*s*Log[r*s] * Derivative[0, 1][QPochhammer][ r*s, r*s])) / (r*s*QPochhammer[r*s, r*s]) == 0}, {r, 1/58}, {s, 4}, WorkingPrecision -> 70] (* Vaclav Kotesovec, Jan 19 2024 *)
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PROG
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(PARI) {a(n) = my(A=[3]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff(2 + sum(m=0, sqrtint(2*#A+9), (-1)^m * x^(m*(m-1)/2) * (1 - x^(2*m+1)) * Ser(A)^(m*(m+1)/2) ), #A-1) ); H=A; A[n+1]}
for(n=0, 30, 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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