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A262718 a(n) = (n+1)^n - 2*(n^n) + (n-1)^n.

%I A262718 #24 Sep 08 2022 08:46:14
%S A262718 0,0,2,18,194,2550,39962,730002,15257090,359376750,9424209002,
%T A262718 272385029466,8604312602690,294957765448710,10906288759973882,
%U A262718 432701819402940450,18336112083960655874,826578941145375829470,39497618599385891373002,1994276034034710498109674
%N A262718 a(n) = (n+1)^n - 2*(n^n) + (n-1)^n.
%C A262718 Obviously, a(n) is always an even number. - _Altug Alkan_, Sep 28 2015
%F A262718 E.g.f.: A(x) = B'(x)*(1-x/B(x))^2, where B(x) is g.f. of A000169.
%F A262718 a(n) = Sum{k=1..n} (k!*binomial(n-1,k-2)*stirling2(n,k)), n>0, a(0)=0.
%F A262718 a(n) = 2*(A062024(n) - A000312(n)). - _Michel Marcus_, Sep 28 2015
%t A262718 Join[{0}, Table[(n + 1)^n - 2 (n^n) + (n - 1)^n, {n, 30}]] (* _Vincenzo Librandi_, Sep 28 2015 *)
%o A262718 (Maxima)
%o A262718 B(x):=-lambert_w(-x);
%o A262718 makelist(n!*coeff(taylor(diff(B(x),x)*(1-x/B(x))^2,x,0,20),x,n),n,0,10);
%o A262718 (PARI) a(n) = (n+1)^n - 2*(n^n) + (n-1)^n;
%o A262718 vector(30, n, a(n-1)) \\ _Altug Alkan_, Sep 28 2015
%o A262718 (Magma) [(n+1)^n - 2*(n^n) + (n-1)^n: n in [0..30]]; // _Vincenzo Librandi_, Sep 28 2015
%Y A262718 Cf. A000169, A000312, A062024.
%K A262718 nonn,easy
%O A262718 0,3
%A A262718 _Vladimir Kruchinin_, Sep 28 2015