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%I #7 May 10 2021 07:38:32
%S 1,1,-3,1,73,-699,3001,24697,-783999,10946233,-80958779,-656003919,
%T 40097528857,-944102982419,14449693290033,-81180376526759,
%U -4110744092532479,203618771909117937,-5868277577182238579,117997016943575159713,-1055340561026036009559,-45279878749358024400299
%N a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*|Lah(n, k)|. Inverse binomial convolution of the unsigned Lah numbers A271703.
%F a(n) = (-1)^(n-1)*n*n!*hypergeom([1 - n, 1 - n], [2, 2], -1) for n >= 1.
%p aList := proc(len) local lah;
%p lah := (n, k) -> `if`(n = k, 1, binomial(n-1, k-1)*n!/k!):
%p seq(add((-1)^(n-k)*binomial(n, k)*lah(n, k), k = 0..n), n = 0..len-1) end:
%p print( aList(22) );
%t a[n_] := (-1)^(n-1) n n! HypergeometricPFQ[{1 - n, 1 - n}, {2, 2}, -1]; a[0] := 1;
%t Table[a[n], {n, 0, 20}]
%Y Cf. A271703, A111596, A344050.
%K sign
%O 0,3
%A _Peter Luschny_, May 10 2021