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A199576
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E.g.f. satisfies: A(x) = 1 + Sum_{n>=1} 2*cosh(n*x) * (x*A(x))^(n^2).
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2
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1, 2, 8, 54, 528, 7690, 159360, 4158014, 124676608, 4135694994, 150596213760, 6039479167942, 267570859708416, 13065427748191130, 697795440530685952, 40363581479632997070, 2506569447568159211520, 166039728985388864259874, 11683091140202632175419392
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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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E.g.f. satisfies: A(x) = Product_{n>=1} (1 - (x*A(x))^(2*n))*(1 + (x*A(x))^(2*n-1)*exp(x))*(1 + (x*A(x))^(2*n-1)/exp(x)), due to the Jacobi triple product identity.
a(n) ~ c * d^n * n! / n^(3/2), where d = 4.245909... and c = 0.72689... - Vaclav Kotesovec, Sep 04 2017
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EXAMPLE
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E.g.f.: A(x) = 1 + 2*x + 8*x^2/2! + 54*x^3/3! + 528*x^4/4! + 7690*x^5/5! +...
The e.g.f. A = A(x) may be expressed by the series:
A(x) = 1 + 2*cosh(x)*x*A + 2*cosh(2*x)*(x*A)^4 + 2*cosh(3*x)*(x*A)^9 + 2*cosh(4*x)*(x*A)^16 + 2*cosh(5*x)*(x*A)^25 +...
and by the Jacobi triple product:
A(x) = (1-(x*A)^2)*(1+x*A*exp(x))*(1+x*A/exp(x)) * (1-(x*A)^4)*(1+(x*A)^3*exp(x))*(1+(x*A)^3/exp(x)) * (1-(x*A)^6)*(1+(x*A)^5*exp(x))*(1+(x*A)^5/exp(x)) * (1-(x*A)^8)*(1+(x*A)^7*exp(x))*(1+(x*A)^7/exp(x)) *...
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MATHEMATICA
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(* Calculation of constant d: *) 1/r /. FindRoot[{E^r * r^2*s*QPochhammer[-1/(E^r * r*s), r^2*s^2] * QPochhammer[-E^r / (r*s), r^2*s^2] * QPochhammer[r^2*s^2, r^2*s^2] == (E^r + r*s)*(1 + E^r * r*s), -2*QPolyGamma[0, 1, r^2*s^2] / Log[r*s] + QPolyGamma[0, Log[-1/(E^r * r*s)]/Log[r^2*s^2], r^2*s^2]/ Log[r*s] + QPolyGamma[0, Log[-E^r / (r*s)] / Log[r^2*s^2], r^2*s^2]/Log[r*s] + (4*E^r / ((E^r + r*s)*(1 + E^r * r*s)))*(1 + r*s*Cosh[r] + r^2*s^2*(r^2*s*QPochhammer[-E^r / (r*s), r^2*s^2] * QPochhammer[r^2*s^2, r^2*s^2] * Derivative[0, 1][QPochhammer][-1/(E^r * r*s), r^2*s^2] + ((1 + r^2*s^2 + 2*r*s * Cosh[r]) * Derivative[0, 1][QPochhammer][-E^r / (r*s), r^2*s^2]) / QPochhammer[-E^r / (r*s), r^2*s^2] + ((1 + r^2*s^2 + 2*r*s*Cosh[r]) * Derivative[0, 1][QPochhammer][r^2*s^2, r^2*s^2]) / QPochhammer[r^2*s^2, r^2*s^2])) == 2}, {r, 1/4}, {s, 2}, WorkingPrecision -> 70] (* Vaclav Kotesovec, Jan 19 2024 *)
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PROG
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(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, sqrtint(n+1), 2*cosh(m*x+x*O(x^n))*(x*A)^(m^2))); n!*polcoeff(A, n)}
(PARI) /* By Jacobi's Triple Product Identity: */
{a(n)=local(A=1+x); for(i=1, n, A=prod(m=1, n\2+1, (1-(x*A)^(2*m))*(1+exp(x+x*O(x^n))*(x*A)^(2*m-1))*(1+exp(-x+x*O(x^n))*(x*A)^(2*m-1)+x*O(x^n)))); n!*polcoeff(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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