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E.g.f. A(x) satisfies: Product_{n>=1} (1 - (-A(x))^n) = exp(x).
3

%I #18 Sep 28 2023 12:04:48

%S 1,3,19,207,3331,71223,1890379,59652687,2175761971,89953773543,

%T 4155502117339,212122704251967,11857607972675011,720435277883199063,

%U 47273215180877201899,3331797538738820992047,251025685429022007354451,20133640365773761748643783,1712740622904757368673592059

%N E.g.f. A(x) satisfies: Product_{n>=1} (1 - (-A(x))^n) = exp(x).

%C Unsigned version of A180563.

%H Paul D. Hanna, <a href="/A294330/b294330.txt">Table of n, a(n) for n = 1..300</a>

%F E.g.f. A(x) satisfies:

%F (1) Sum_{n>=1} (-1)^(n-1) * sigma(n) * A(x)^n / n = x.

%F (2) Sum_{n>=0} (-1)^[n/2] * (2*n+1) * A(x)^(n*(n+1)/2) = exp(3*x).

%F (3) A(x) = Series_Reversion( log(Q(x)) ) where Q(x) = Product_{n>=1} (1 - (-x)^n).

%F a(n) ~ c * d^n * n^(n-1), where d = 1.788680223969315995... and c = 0.254472375755339325... - _Vaclav Kotesovec_, Oct 29 2017

%e 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! +...

%e such that A( log(Q(x)) ) = x, where:

%e Q(x) = Product_{n>=1} (1 - (-x)^n);

%e 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 +...

%e 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 +...

%e Also,

%e 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) +...

%e ALTERNATE GENERATING FUNCTION.

%e 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 +...

%e such that

%e 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 +...

%t (* 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 *)

%o (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)}

%o for(n=1, 20, print1(a(n), ", "))

%Y Cf. A294331, A010815, A180563 (variant).

%K nonn

%O 1,2

%A _Paul D. Hanna_, Oct 28 2017