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A337558
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E.g.f. A(x) satisfies: A(x) = Sum_{n>=0} (n+1 - A(x))^n * x^n/n!.
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1
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1, 1, 2, 9, 76, 865, 11976, 197449, 3798992, 83513889, 2063157760, 56576301001, 1705317547968, 56039920728961, 1993964008270976, 76368613495847625, 3132518434553118976, 137006108936644781761, 6364605036943211332608, 312966349102938272936329, 16240099797362530389324800
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OFFSET
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0,3
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LINKS
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FORMULA
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E.g.f. A(x) satisfies:
(1) Sum_{n>=0} (n+1 - A(x))^n * x^n/n! = A(x).
(2) Sum_{n>=0} (n+2 - A(x))^n * x^n/n! = A(x) * LambertW(-x)/(-x).
(3) Sum_{n>=0} (n+m - A(x))^n * x^n/n! = A(x) * ( LambertW(-x)/(-x) )^(m-1).
(4) Sum_{n>=0} (n+1 - 2*A(x))^n * x^n/n! = A(x)^2 * (1 + LambertW(-x)) / ( LambertW(-x)/(-x) ).
(5) Sum_{n>=0} (n+m - p*A(x))^n * x^n/n! = A(x)^p * (1 + LambertW(-x))^(p-1) * ( LambertW(-x)/(-x) )^(m-p).
(6) A(x) = ( LambertW(-x)/(-x) )^(1 - A(x)) / (1 + LambertW(-x)).
(7) A(x) = exp( LambertW(-x)*A(x) ) * ( LambertW(-x)/(-x) ) / (1 + LambertW(-x)).
E.g.f.: -LambertW(LambertW(-x)^2/(x + x*LambertW(-x))) / LambertW(-x). - Vaclav Kotesovec, Sep 01 2020
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EXAMPLE
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E.g.f.: A(x) = 1 + x + 2*x^2/2! + 9*x^3/3! + 76*x^4/4! + 865*x^5/5! + 11976*x^6/6! + 197449*x^7/7! + 3798992*x^8/8! + 83513889*x^9/9! + 2063157760*x^10/10! + ...
where
A(x) = 1 + (2 - A(x))*x + (3 - A(x))^2*x^2/2! + (4 - A(x))^3*x^3/3! + (5 - A(x))^4*x^4/4! + (6 - A(x))^5*x^5/5! + (7 - A(x))^6*x^6/6! + ...
Also,
A(x) * LambertW(-x)/(-x) = 1 + (3 - A(x))*x + (4 - A(x))^2*x^2/2! + (5 - A(x))^3*x^3/3! + (6 - A(x))^4*x^4/4! + (7 - A(x))^5*x^5/5! + ...
where
LambertW(-x)/(-x) = 1 + x + 3*x^2/2! + 16*x^3/3! + 125*x^4/4! + 1296*x^5/5! + 16807*x^6/6! + ... + (n+1)^(n-1)*x^n/n! + ...
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MATHEMATICA
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nmax = 20; CoefficientList[Series[-LambertW[LambertW[-x]^2/(x + x*LambertW[-x])]/LambertW[-x], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Sep 01 2020 *)
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PROG
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(PARI) {a(n) = my(A=1 +O(x)); for(i=1, n, A = sum(m=0, n, (m+1 - A)^m *x^m/m!) ); n!*polcoeff(A, n)}
for(n=0, 30, 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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