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A219619 a(n) = n! * (n^4 + n^2 + 1).

%I A219619 #30 Jan 30 2021 04:18:09
%S A219619 1,3,42,546,6552,78120,959760,12353040,167771520,2410611840,
%T A219619 36654508800,589291718400,10002032409600,178908534604800,
%U A219619 3366215358105600,66496549287168000,1376573115101184000,29810519036153856000,674176353586864128000,15896946656727392256000
%N A219619 a(n) = n! * (n^4 + n^2 + 1).
%C A219619 Sum_{n>=0} 1/a(n) = e/2.
%H A219619 Iain Fox, <a href="/A219619/b219619.txt">Table of n, a(n) for n = 0..445</a>
%H A219619 G. I. Senum, <a href="http://www.jstor.org/stable/2323815">A Series for e, Problem E3352</a>, Amer. Math. Monthly 98 (1991), No. 4, pp. 369-370.
%F A219619 a(n) = A000142(n)*A059826(n). - _Michel Marcus_, Nov 19 2017
%F A219619 E.g.f.: (-3*x^4 - 6*x^3 - 16*x^2 + 2*x - 1)/(x - 1)^5. - _Iain Fox_, Nov 19 2017
%e A219619 a(3) = 3!*(3^4 + 3^2 + 1) = 6*91 = 546.
%t A219619 Array[#!*(#^4 + #^2 + 1) &, 20, 0] (* _Michael De Vlieger_, Jan 29 2021 *)
%t A219619 nmax = 20; CoefficientList[Series[(1 - 2*x + 16*x^2 + 6*x^3 + 3*x^4) / (1 - x)^5, {x, 0, nmax}], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, Jan 29 2021 *)
%o A219619 (PARI) a(n) = n! * (n^4 + n^2 + 1); \\ _Michel Marcus_, Nov 19 2017
%o A219619 (PARI) first(n) = { my(x='x+O('x^n)); Vec(serlaplace((-3*x^4-6*x^3-16*x^2+2*x-1)/(x-1)^5)); } \\ _Iain Fox_, Nov 19 2017
%Y A219619 Cf. A000142 (n!), A059826 (n^4 + n^2 + 1).
%K A219619 nonn,easy
%O A219619 0,2
%A A219619 _Franz Vrabec_, Nov 24 2012