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%I #11 Feb 18 2018 15:54:46
%S 1,1,4,39,688,18765,722016,37003267,2426725120,197569640889,
%T 19498786969600,2288864397602871,314642887620065280,
%U 50002239358837749061,9086161251793519280128,1870292375864728408312875,432549223770637009573052416,111598253780454986901562293489,31918043775392401233962828169216,10063176613379061071841096012386911,3479934178969181147698311417364480000
%N E.g.f. Sum_{n>=0} Series_Reversion( x*exp(-n*x) )^n.
%H Paul D. Hanna, <a href="/A299426/b299426.txt">Table of n, a(n) for n = 0..400</a>
%F E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following expressions.
%F (1) A(x) = Sum_{n>=0} Series_Reversion( x*exp(-n*x) )^n.
%F (2) A(x) = Sum_{n>=1} x^n * W(x,n)^(n^2), where
%F (2.a) W(x,n) = exp( -LambertW(-n*x)/n ),
%F (2.b) W(x,n) = exp( x*W(x,n)^n ),
%F (2.c) W(x,n)^n = Series_Reversion( x*exp(-n*x) ) / x,
%F (2.d) W(x,n)^n = Sum_{k>=0} n^k * (k+1)^(k-1) * x^k/k!;
%F (2.e) W(x,n)^(n^2) = Sum_{k>=0} n^(k+1) * (n+k)^(k-1) * x^k/k!.
%F FORMULAS FOR TERMS.
%F a(n) = Sum_{k=0..n} n!/k! * n^(k-1) * (n-k)^(k+1).
%e E.g.f. A(x) = 1 + x + 4*x^2/2! + 39*x^3/3! + 688*x^4/4! + 18765*x^5/5! + 722016*x^6/6! + 37003267*x^7/7! + 2426725120*x^8/8! + ...
%e such that
%e A(x) = 1 + x*W(x,1) + x^2*W(x,2)^4 + x^3*W(x,3)^9 + x^4*W(x,4)^16 + x^5*W(x,5)^25 + x^6*W(x,6)^36 + ...
%e where series W(x,n) = exp( x*W(x,n)^n ) begin:
%e W(x,1) = 1 + x + 3*x^2/2! + 16*x^3/3! + 125*x^4/4! + 1296*x^5/5! + ...
%e W(x,2) = 1 + x + 5*x^2/2! + 49*x^3/3! + 729*x^4/4! + 14641*x^5/5! + ...
%e W(x,3) = 1 + x + 7*x^2/2! + 100*x^3/3! + 2197*x^4/4! + 65536*x^5/5! + ...
%e W(x,4) = 1 + x + 9*x^2/2! + 169*x^3/3! + 4913*x^4/4! + 194481*x^5/5! + ...
%e W(x,5) = 1 + x + 11*x^2/2! + 256*x^3/3! + 9261*x^4/4! + 456976*x^5/5! + ...
%e W(x,6) = 1 + x + 13*x^2/2! + 361*x^3/3! + 15625*x^4/4! + 923521*x^5/5! + ...
%e ...
%e and series W(x,n)^n = Series_Reversion( x*exp(-n*x) ) / x begin:
%e W(x,1) = 1 + x + 3*x^2/2! + 16*x^3/3! + 125*x^4/4! + 1296*x^5/5! + ...
%e W(x,2)^2 = 1 + 2*x + 12*x^2/2! + 128*x^3/3! + 2000*x^4/4! + ...
%e W(x,3)^3 = 1 + 3*x + 27*x^2/2! + 432*x^3/3! + 10125*x^4/4! + ...
%e W(x,4)^4 = 1 + 4*x + 48*x^2/2! + 1024*x^3/3! + 32000*x^4/4! + ...
%e W(x,5)^5 = 1 + 5*x + 75*x^2/2! + 2000*x^3/3! + 78125*x^4/4! + ...
%e W(x,6)^6 = 1 + 6*x + 108*x^2/2! + 3456*x^3/3! + 162000*x^4/4! + ...
%e ...
%e and series W(x,n)^(n^2) begin:
%e W(x,1) = 1 + x + 3*x^2/2! + 16*x^3/3! + 125*x^4/4! + 1296*x^5/5! + ...
%e W(x,2)^4 = 1 + 4*x + 32*x^2/2! + 400*x^3/3! + 6912*x^4/4! + ...
%e W(x,3)^9 = 1 + 9*x + 135*x^2/2! + 2916*x^3/3! + 83349*x^4/4! + ...
%e W(x,4)^16 = 1 + 16*x + 384*x^2/2! + 12544*x^3/3! + 524288*x^4/4! + ...
%e W(x,5)^25 = 1 + 25*x + 875*x^2/2! + 40000*x^3/3! + 2278125*x^4/4! + ...
%e W(x,6)^36 = 1 + 36*x + 1728*x^2/2! + 104976*x^3/3! + 7776000*x^4/4! + ...
%e ...
%o (PARI) {a(n) = my(A); A = sum(m=0,n+1, serreverse( x*exp(-m*x +x^2*O(x^n) ) )^m ); n!*polcoeff(A,n)}
%o for(n=0,20,print1(a(n),", "))
%o (PARI) {a(n) = if(n==0,1, sum(k=0,n, n!/k! * n^(k-1) * (n-k)^(k+1) ) )}
%o for(n=0,20,print1(a(n),", "))
%Y Cf. A244468, A299044.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Feb 18 2018
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