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%I #11 Jun 25 2016 13:04:18
%S 1,2,9,64,595,7416,111979,1989632,40695561,941667040,24323649361,
%T 693818707968,21661372820971,734712173277824,26902827107293635,
%U 1057724890214957056,44442356900221356241,1987370544970750468608,94240073170115929379161,4723448516579307027169280,249510355552473169494452931,13854414947224528743034304512,806733172355775780726416256859
%N E.g.f. A(x) satisfies: A( A( x^4*exp(-4*x) )^(1/4) ) = x.
%H Paul D. Hanna, <a href="/A274394/b274394.txt">Table of n, a(n) for n = 1..100</a>
%F E.g.f. A(x) = Sum_{n>=1} a(n) * x^n / n! satisfies:
%F (1) A( A( x^4*exp(4*x) )^(1/4) ) = -LambertW(-x*exp(x)).
%F (2) A(x) = Series_Reversion( A( x^4*exp(-4*x) )^(1/4) ).
%F (3) A( A(x)^4 * exp(-4*A(x)) ) = x^4.
%F (4) A(-A(x)^4 * exp(-4*A(x)) ) = -LambertW(x^4*exp(-x^4)).
%e E.g.f.: A(x) = x + 2*x^2/2! + 9*x^3/3! + 64*x^4/4! + 595*x^5/5! + 7416*x^6/6! + 111979*x^7/7! + 1989632*x^8/8! + 40695561*x^9/9! + 941667040*x^10/10! + 24323649361*x^11/11! + 693818707968*x^12/12! + 21661372820971*x^13/13! + 734712173277824*x^14/14! + 26902827107293635*x^15/15! + 1057724890214957056*x^16/16! +...
%e such that A( A( x^4*exp(-4*x) )^(1/4) ) = x.
%e RELATED SERIES.
%e The series reversion of the e.g.f. A(x) equals the series defined by:
%e A( x^4*exp(-4*x) )^(1/4) = x - 2*x^2/2! + 3*x^3/3! - 4*x^4/4! + 35*x^5/5! - 906*x^6/6! + 15757*x^7/7! - 210008*x^8/8! + 2464569*x^9/9! - 32810410*x^10/10! + 671239811*x^11/11! - 18224632812*x^12/12! + 496597765963*x^13/13! - 12681217528994*x^14/14! + 320976165059565*x^15/15! +...
%e Compare the above series reversion to the following series:
%e A(x)^4 * exp(-4*A(x)) = x^4 - 2*x^8/2! + 3*x^12/3! - 4*x^16/4! + 35*x^20/5! - 906*x^24/6! + 15757*x^28/7! - 210008*x^32/8! + 2464569*x^36/9! - 32810410*x^40/10! +...
%e where A( A(x)^4 * exp(-4*A(x)) ) = x^4.
%e The e.g.f. A(x) is related to the LambertW function by the composition:
%e A( A(x^4*exp(4*x))^(1/4) ) = x + 4*x^2/2! + 24*x^3/3! + 224*x^4/4! + 2880*x^5/5! + 47232*x^6/6! + 942592*x^7/7! + 22171648*x^8/8! +...+ A216857(n)*x^n/n! +...
%e which equals -LambertW(-x*exp(x)).
%o (PARI) {a(n) = my(A=x); for(i=1, n, A = serreverse( subst(A, x, x^4*exp(-4*x +x*O(x^n)))^(1/4) ) ); n!*polcoeff(A, n)}
%o for(n=1, 30, print1(a(n), ", "))
%Y Cf. A274275, A274393, A274395.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Jun 21 2016
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