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A294330
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E.g.f. A(x) satisfies: Product_{n>=1} (1 - (-A(x))^n) = exp(x).
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3
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1, 3, 19, 207, 3331, 71223, 1890379, 59652687, 2175761971, 89953773543, 4155502117339, 212122704251967, 11857607972675011, 720435277883199063, 47273215180877201899, 3331797538738820992047, 251025685429022007354451, 20133640365773761748643783, 1712740622904757368673592059
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OFFSET
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1,2
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COMMENTS
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LINKS
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FORMULA
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E.g.f. A(x) satisfies:
(1) Sum_{n>=1} (-1)^(n-1) * sigma(n) * A(x)^n / n = x.
(2) Sum_{n>=0} (-1)^[n/2] * (2*n+1) * A(x)^(n*(n+1)/2) = exp(3*x).
(3) A(x) = Series_Reversion( log(Q(x)) ) where Q(x) = Product_{n>=1} (1 - (-x)^n).
a(n) ~ c * d^n * n^(n-1), where d = 1.788680223969315995... and c = 0.254472375755339325... - Vaclav Kotesovec, Oct 29 2017
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EXAMPLE
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E.g.f.: A(x) = x + 3*x^2/2! + 19*x^3/3! + 207*x^4/4! + 3331*x^5/5! + 71223*x^6/6! + 1890379*x^7/7! + 59652687*x^8/8! + 2175761971*x^9/9! + 89953773543*x^10/10! +...
such that A( log(Q(x)) ) = x, where:
Q(x) = Product_{n>=1} (1 - (-x)^n);
log(Q(x)) = x - 3*x^2/2 + 4*x^3/3 - 7*x^4/4 + 6*x^5/5 - 12*x^6/6 + 8*x^7/7 - 15*x^8/8 + 13*x^9/9 - 18*x^10/10 +...+ (-1)^(n-1)*sigma(n)*x^n/n +...
and Q(x) = 1 + x - x^2 - x^5 - x^7 - x^12 + x^15 + x^22 + x^26 + x^35 - x^40 - x^51 - x^57 - x^70 + x^77 + x^92 + x^100 +...+ A121373(n)*x^n +...
Also,
exp(3*x) = 1 + 3*A(x) - 5*A(x)^3 - 7*A(x)^6 + 9*A(x)^10 + 11*A(x)^15 - 13*A(x)^21 - 15*A(x)^28 + 17*A(x)^36 +...+ (-1)^[n/2] * (2*n+1) * A(x)^(n*(n+1)/2) +...
ALTERNATE GENERATING FUNCTION.
L.g.f.: L(x) = x + 3*x^2/2 + 19*x^3/3 + 207*x^4/4 + 3331*x^5/5 + 71223*x^6/6 + 1890379*x^7/7 + 59652687*x^8/8 + 2175761971*x^9/9 + 89953773543*x^10/10 +...
such that
exp(L(x)) = 1 + x + 2*x^2 + 8*x^3 + 60*x^4 + 732*x^5 + 12672*x^6 + 283704*x^7 + 7757526*x^8 + 249885110*x^9 + 9255184676*x^10 +...+ A294331(n)*x^n +...
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MATHEMATICA
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(* Calculation of constants {d, c}: *) eq = FindRoot[{E^r == QPochhammer[-s], (E^r*(Log[1 + s] + QPolyGamma[0, 1, -s]))/(s*Log[-s]) + Derivative[0, 1][QPochhammer][-s, -s] == 0}, {r, 1/5}, {s, 1/2}, WorkingPrecision -> 400]; {N[1/r/E /. eq, 120], val = s*E^r*Sqrt[-r*(1 + s) * (Log[-s]^2/(E^(2*r)*(1 + s)*QPolyGamma[1, 1, -s] + s*Log[-s]*(-s*(1 + s) * Log[-s] * Derivative[0, 1][QPochhammer][-s, -s]^2 + E^r*(1 + s)*((-2 - Log[-s]) * Derivative[0, 1][QPochhammer][-s, -s] + s*Log[-s] * Derivative[0, 2][QPochhammer][-s, -s]) + 2*E^(2*r)*(-1 + (1 + s) * Derivative[0, 0, 1][QPolyGamma][0, 1, -s]))))] /. eq; N[Chop[val], -Floor[Log[10, Abs[Im[val]]]] - 3]} (* Vaclav Kotesovec, Sep 28 2023 *)
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PROG
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(PARI) {a(n) = local( L = sum(m=1, n, (-1)^(m-1) * sigma(m) * x^m/m ) +x*O(x^n) ); n!*polcoeff( serreverse(L), n)}
for(n=1, 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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