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A243954 E.g.f.: A(x) = x/Series_Reversion(x*G(x)) where G(x) = Sum_{n>=0} n^n*x^n/n!.

%I A243954 #5 Jun 20 2014 17:48:33
%S A243954 1,1,2,3,-8,-95,-144,5299,51584,-263007,-11283200,-61279229,
%T A243954 2321353728,49471727617,-217783810048,-25824369466125,
%U A243954 -298783278792704,10591512134358721,400315590893961216,-536860171776183677,-362306852971970560000,-7292410691309793586719,227379942363338919903232
%N A243954 E.g.f.: A(x) = x/Series_Reversion(x*G(x)) where G(x) = Sum_{n>=0} n^n*x^n/n!.
%C A243954 Compare to: exp(x) = x/Series_Reversion(x*W(x)) where W(x) = Sum_{n>=0} (n+1)^(n-1)*x^n/n!.
%H A243954 Paul D. Hanna, <a href="/A243954/b243954.txt">Table of n, a(n) for n = 0..100</a>
%F A243954 E.g.f. satisfies: A(x) = G(x/A(x)) such that A(x*G(x)) = G(x) = Sum_{n>=0} n^n*x^n/n!.
%F A243954 E.g.f. A(x) satisfies: [x^n/n!] A(x)^(n+1)/(n+1) = n^n.
%e A243954 G.f.: A(x) = 1 + x + 2*x^2/2! + 3*x^3/3! - 8*x^4/4! - 95*x^5/5! - 144*x^6/6! +...
%e A243954 where
%e A243954 A(x) = 1 + x/A(x) + 4*(x/A(x))^2/2! + 27*(x/A(x))^3/3! + 256*(x/A(x))^4/4! + 3125*(x/A(x))^5/5! +...+ n^n*(x/A(x))^n/n! +...
%e A243954 If we form a table of coefficients of x^k/k! in A(x)^n, like so:
%e A243954 [1, 1,  2,   3,    -8,    -95,    -144,     5299,     51584, ...];
%e A243954 [1, 2,  6,  18,    32,   -150,   -1728,    -1078,    144384, ...];
%e A243954 [1, 3, 12,  51,   192,    375,   -2592,   -29841,     12288, ...];
%e A243954 [1, 4, 20, 108,   568,   2500,    5184,   -48020,   -557056, ...];
%e A243954 [1, 5, 30, 195,  1280,   7845,   38880,    84035,   -983040, ...];
%e A243954 [1, 6, 42, 318,  2472,  18750,  129456,   705894,   1572864, ...];
%e A243954 [1, 7, 56, 483,  4312,  38395,  326592,  2485567,  14680064, ...];
%e A243954 [1, 8, 72, 696,  6992,  70920,  704448,  6588344,  54442368, ...];
%e A243954 [1, 9, 90, 963, 10728, 121545, 1368144, 14890995, 150994944, ...]; ...
%e A243954 then the main diagonal equals (k+1)*k^k for k>=0.
%o A243954 (PARI) {a(n)=n!*polcoeff(x/serreverse(x*sum(m=0, n+1, m^m*x^m/m!)+x^2*O(x^n)), n)}
%o A243954 for(n=0,25,print1(a(n),", "))
%K A243954 sign
%O A243954 0,3
%A A243954 _Paul D. Hanna_, Jun 20 2014