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%I #8 Jul 05 2018 07:22:23
%S 1,5,27,142,847,5817,45733,405836,4012701,43733965,520794991,
%T 6726601050,93651619867,1398047697137,22275111534537,377278848390232,
%U 6768744159489913,128228860181918421,2557808459478878851,53585748788874537830,1176328664895760953831
%N a(n) = Sum_{k=1..n} (-1)^(n-k)*hypergeom([k, k-2-n], [], 1).
%F a(n) = A292898(n, 2).
%F From _Vaclav Kotesovec_, Jul 05 2018: (Start)
%F Recurrence: (n^2 - 4*n + 5)*a(n) = (n^3 - 3*n^2 + 3*n + 2)*a(n-1) - (n-1)*(2*n - 3)*a(n-2) - (n^3 - 3*n^2 + 2*n + 1)*a(n-3) + (n^2 - 2*n + 2)*a(n-4).
%F a(n) ~ n * n!.
%F a(n) ~ sqrt(2*Pi) * n^(n + 3/2) / exp(n). (End)
%p A293295 := n -> add((-1)^(n-k)*hypergeom([k, k-2-n], [], 1), k=1..n):
%p seq(simplify(A293295(n)), n=1..20);
%t Table[Sum[(-1)^(n-k)*HypergeometricPFQ[{k, k-2-n}, {}, 1], {k,1,n}], {n,1,20}] (* _Vaclav Kotesovec_, Jul 05 2018 *)
%Y Cf. A003470 (n=0), A193464 (n=1), this sequence (n=2), A292898 (n>=0).
%K nonn
%O 1,2
%A _Peter Luschny_, Oct 05 2017
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