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A322943 a(n) = n! [x^n] -exp(-1/(3*(x - 1)^3) - 1/3)/(x - 1).

%I A322943 #14 Mar 24 2020 04:41:57
%S A322943 1,2,9,60,513,5286,63417,865824,13229505,223336458,4123226601,
%T A322943 82559530692,1780580892609,41125146159150,1012187976013593,
%U A322943 26434618529133096,729843662368002177,21233024209964157714,649022529915336217545,20789723945673232443468,696253958136289126229121
%N A322943 a(n) = n! [x^n] -exp(-1/(3*(x - 1)^3) - 1/3)/(x - 1).
%F A322943 a(n) = (4*n - 2)*a(n-1) - 3*(n - 1)*(2*n - 3)*a(n-2) + (n - 1)*(n - 2)*(4*n - 9)*a(n-3) - (n - 2)*(n - 1)*(n - 3)^2*a(n-4) for n >= 4.
%F A322943 a(n) ~ exp(-1/4 + 5*n^(1/4)/24 + sqrt(n)/2 + 4*n^(3/4)/3 - n) * n^(n + 1/8) / 2 * (1 + 1357/(2560*n^(1/4))). - _Vaclav Kotesovec_, Jan 02 2019
%p A322943 a := proc(n) option remember; local e, b, c, d;
%p A322943 e := n -> 4*n-2; b := n -> 3*(n-1)*(2*n-3);
%p A322943 c := n -> (n-1)*(n-2)*(4*n-9); d := n -> (n-2)*(n-1)*(n-3)^2;
%p A322943 if n < 4 then return [1, 2, 9, 60][n+1] fi;
%p A322943 e(n)*a(n-1) - b(n)*a(n-2) + c(n)*a(n-3) - d(n)*a(n-4) end:
%p A322943 seq(a(n), n=0..20);
%o A322943 (Sage) # uses[riordan_square from A321620]
%o A322943 R = riordan_square((1 - 3*x)^(-1/3), 24, True).inverse()
%o A322943 [sum([(-1)^(n-k)*c for k, c in enumerate(R.row(n))]) for n in (0..23)]
%Y A322943 Row sums of A322944.
%Y A322943 Cf. A322944, A321620, A321965.
%K A322943 nonn
%O A322943 0,2
%A A322943 _Peter Luschny_, Jan 02 2019