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%I #15 Nov 28 2021 13:41:07
%S 1,1,-1,-5,13,101,-389,-4241,21785,305353,-1960649,-33588829,
%T 258805669,5239857325,-47102631757,-1100370038249,11304631621681,
%U 299300650403729,-3459217276234385,-102360822438075317,1314502564969066301,42991545423991633141,-607300185015708631061
%N a(n) = n! * Sum_{k=0..floor(n/2)} (-1)^k / (n-2*k)!.
%H Seiichi Manyama, <a href="/A337749/b337749.txt">Table of n, a(n) for n = 0..449</a>
%F G.f.: Sum_{k>=0} (-1)^k * (2*k)! * x^(2*k) / (1 - x)^(2*k+1).
%F E.g.f.: exp(x) / (1 + x^2).
%F a(0) = a(1) = 1; a(n) = 1 - n * (n-1) * a(n-2).
%t Table[n! Sum[(-1)^k/(n - 2 k)!, {k, 0, Floor[n/2]}], {n, 0, 22}]
%t nmax = 22; CoefficientList[Series[Exp[x]/(1 + x^2), {x, 0, nmax}], x] Range[0, nmax]!
%t (* alternative code *)
%t f[x_]:=I*(ExpIntegralE[-x,I]*E^I-ExpIntegralE[-x,-I]*E^(-I))/2
%t FunctionExpand[Array[f,20,0]] (* _Velin Yanev_, Oct 13 2021 *)
%o (PARI) a(n) = n!*sum(k=0, n\2, (-1)^k / (n-2*k)!); \\ _Michel Marcus_, Sep 18 2020
%Y Cf. A087208, A182386, A337750, A337751.
%K sign
%O 0,4
%A _Ilya Gutkovskiy_, Sep 18 2020
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