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A268651
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G.f. A(x) satisfies: 1 = Product_{n>=1} (1 - x^n) * (1 - x^(n+1)/A(x)) * (1 - x^(n-2)*A(x)).
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2
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1, 1, 2, 5, 9, 22, 52, 146, 377, 1036, 2810, 8014, 22790, 66100, 191541, 562926, 1660975, 4944766, 14767136, 44357952, 133698623, 404810569, 1229434572, 3746595869, 11447723074, 35075829156, 107724187826, 331605018200, 1022842337041, 3161156987190, 9787096605716, 30352665554591, 94279407445012, 293277650593792, 913565090912339, 2849489942324466, 8898714901181309, 27822251614174021, 87083081436755770
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OFFSET
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0,3
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COMMENTS
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The g.f. utilizes the Jacobi Triple Product:
Product_{n>=1} (1-x^n)*(1 - x^n/a)*(1 - x^(n-1)*a) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)/2) * a^n.
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LINKS
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FORMULA
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G.f. A(x) satisfies:
(1) x = Sum_{n=-oo..oo} (-1)^n * x^((n-1)*(n-2)/2) * A(x)^n.
(2) x = sum_{n>=0} (-1)^n * x^(n*(n+1)/2) * (A(x)^(2*n+1) - 1) / A(x)^(n-1).
(3) A(x) = x / Series_Reversion( G(x) ), where G(x) is the g.f. of A268650.
a(n) ~ c * d^n / n^(3/2), where d = 3.25766000970998791... and c = 0.661369655158037... . - Vaclav Kotesovec, Mar 05 2016
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EXAMPLE
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G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 9*x^4 + 22*x^5 + 52*x^6 + 146*x^7 + 377*x^8 + 1036*x^9 + 2810*x^10 + 8014*x^11 + 22790*x^12 + 66100*x^13 + 191541*x^14 + 562926*x^15 +...
where A(x) satisfies the Jacobi Triple Product:
1 = (1-x)*(1-x^2/A(x))*(1-1/x*A(x)) * (1-x^2)*(1-x^3/A(x))*(1-1*A(x)) * (1-x^3)*(1-x^4/A(x))*(1-x*A(x)) * (1-x^4)*(1-x^5/A(x))*(1-x^2*A(x)) * (1-x^5)*(1-x^6/A(x))*(1-x^3*A(x)) * (1-x^6)*(1-x^7/A(x))*(1-x^4*A(x)) *...
Also
x = (A(x)-1)*A(x) - x*(A(x)^3-1) + x^3*(A(x)^5-1)/A(x) - x^6*(A(x)^7-1)/A(x)^2 + x^10*(A(x)^9-1)/A(x)^3 - x^15*(A(x)^11-1)/A(x)^4 + x^21*(A(x)^13-1)/A(x)^5 +...
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MATHEMATICA
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(* Calculation of constant d: *) 1/r /. FindRoot[{s*QPochhammer[r, r] * QPochhammer[r/s, r] * QPochhammer[s/r^2, r] == (s - r)*(1 - s/r^2), (r^3 - s^2)* Log[r] + (r^3 - r*s - r^2*s + s^2) * (QPolyGamma[0, Log[r/s]/Log[r], r] - QPolyGamma[0, Log[s/r^2]/Log[r], r]) == 0}, {r, 1/3}, {s, 2}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Sep 30 2023 *)
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PROG
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(PARI) {a(n) = my(A=[1, 1]); for(i=1, n, A=concat(A, 0); A[#A]=-Vec( sum(m=1, sqrtint(2*#A)+2, (-1)^m*(x*Ser(A))^(m*(m-1)/2)*(1-x^(2*m-1))/x^m) )[#A-1] ); Vec(x/serreverse(x*Ser(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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