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%I #23 Sep 08 2022 08:46:02
%S 1,-2,9,-64,625,-7776,117649,-2097152,43046721,-1000000000,
%T 25937424601,-743008370688,23298085122481,-793714773254144,
%U 29192926025390625,-1152921504606846976,48661191875666868481,-2185911559738696531968,104127350297911241532841
%N a(n) = (-n)^(n-1).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Lambert_W_function">Lambert W function</a>
%F E.g.f.: LambertW(x).
%F E.g.f. is the functional inverse of x * exp(x) which is the e.g.f. of A000027.
%F E.g.f. A(x) satisfies A(x) = x / exp(A(x)).
%F a(n) = -(-1)^n * A000169(n).
%e x - 2*x^2 + 9*x^3 - 64*x^4 + 625*x^5 - 7776*x^6 + 117649*x^7 + ...
%p a := proc(n); `if`( n<0, 0, n! * coeff( taylor( LambertW(x), x=0, n+1 ), x, n)); end;
%t a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ ProductLog @ z, {z, 0, n}]]
%t Table[(-n)^(n-1),{n,30}] (* _Harvey P. Dale_, Apr 29 2013 *)
%o (PARI) {a(n) = if( n<1, 0, (-n) ^ (n-1))}
%o (PARI) {a(n) = if( n<1, 0, n! * polcoeff( serreverse( x * exp(x + x * O(x^n))), n))}
%o (PARI) {a(n) = local(A); if( n<1, 0, A = O(x); for( k=1, n, A = x / exp(A)); n! * polcoeff( A, n))}
%o (Magma) [(-n)^(n-1) : n in [1..20]]; // _Wesley Ivan Hurt_, Jan 17 2017
%Y Cf. A000027, A000169.
%K sign
%O 1,2
%A _Michael Somos_, Jun 07 2012
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