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A355349
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G.f. A(x) satisfies: 2 = Sum_{n=-oo..+oo} (-x)^(n*(n-1)/2) * A(x)^(n^2).
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2
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1, 2, 10, 76, 678, 6608, 68170, 731638, 8084692, 91361298, 1050937008, 12264790410, 144856757032, 1728197200206, 20796217437806, 252117655811806, 3076371017010508, 37753163861001044, 465657991700212170, 5769586313420410060, 71777257553636752194
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OFFSET
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0,2
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LINKS
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FORMULA
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G.f. A(x) satisfies:
(1) 2 = Sum_{n=-oo..+oo} (-x)^(n*(n-1)/2) * A(x)^(n^2).
(2) 2 = Product_{n>=1} (1 - (-x)^n*A(x)^(2*n)) * (1 + (-x)^(n-1)*A(x)^(2*n-1)) * (1 + (-x)^n*A(x)^(2*n-1)), by the Jacobi triple product identity.
Formula (2) can be rewritten as the functional equation QPochhammer(-x*y^2) * QPochhammer(1/(x*y), -x*y^2)/(1 - 1/(x*y)) * QPochhammer(-1/y, -x*y^2)/(1 + 1/y) = 2.
a(n) ~ c * d^n / n^(3/2), where d = 13.4235502463317299100709807099986120871056759637108569... and c = 0.1810118197770993236368884418746617625188562698029... (End)
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EXAMPLE
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G.f.: A(x) = 1 + 2*x + 10*x^2 + 76*x^3 + 678*x^4 + 6608*x^5 + 68170*x^6 + 731638*x^7 + 8084692*x^8 + 91361298*x^9 + 1050937008*x^10 + ...
where
2 = ... + x^6*A(x)^9 - x^3*A(x)^4 - x*A(x) + A(x) + 1 - x*A(x) - x^3*A(x)^4 + x^6*A(x)^9 + x^10*A(x)^16 - x^15*A(x)^25 - x^21*A(x)^36 ++-- ... (-x)^(n*(n-1)/2) * A(x)^(n^2) + ...
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MATHEMATICA
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(* Calculation of constant d: *) 1/r /. FindRoot[{r*s^2*QPochhammer[-1/s, -r*s^2]* QPochhammer[1/(r*s), -r*s^2]* QPochhammer[-r*s^2]/((1 + s)*(-1 + r*s)) == 2, (4 + 2*s - 2*r*s)/(r*s^2) + 2*r*s^2*QPochhammer[1/(r*s), -r*s^2]* QPochhammer[-r*s^2] * Derivative[0, 1][QPochhammer][-1/s, -r*s^2] + 4*(1 + s)*(-1 + r*s)* Derivative[0, 1][QPochhammer][1/(r*s), -r*s^2] / QPochhammer[1/(r*s), -r*s^2] + 2*(1 + s)*(-1 + r*s) * ((2*QPolyGamma[0, 1, -r*s^2] - QPolyGamma[0, Log[-1/s]/Log[-r*s^2], -r*s^2] - QPolyGamma[0, Log[1/(r*s)]/Log[-r*s^2], -r*s^2])/(r*s^2* Log[-r*s^2]) + (2*Derivative[0, 1][QPochhammer][-r*s^2, -r*s^2])/ QPochhammer[-r*s^2]) == 0}, {r, 1/12}, {s, 3/2}, WorkingPrecision -> 70] (* Vaclav Kotesovec, Jan 31 2024 *)
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PROG
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(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); M=ceil(sqrt(2*n));
A[#A] = -polcoeff( sum(m=-M, M, (-x)^(m*(m-1)/2) * Ser(A)^(m^2)), #A-1)); A[n+1]}
for(n=0, 20, 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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