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%I #11 Jun 06 2018 19:54:45
%S 1,3,26,429,11140,413575,20732442,1349324599,110687183288,
%T 11178507440925,1363390073347730,197621842687876673,
%U 33583706596253821788,6613771286589113270611,1494257702398724584231946,383919965020249799260370355,111309986933239661174592643760,36168040052169465186266712298305,13090361918944129697134824216839154,5248357548475200108500930014786066693,2319429294524000405318027925169538777460
%N O.g.f. A(x) satisfies: [x^n] exp( n * x*A(x) ) * (n + 3 - A(x)) = 0 for n > 0.
%C Note: the factorial series, F(x) = Sum_{n>=0} n! * x^n, satisfies:
%C (1) [x^n] exp( n * x*F(x) ) * (2 - F(x)) = 0 for n > 0,
%C (2) [x^n] exp( x*F(x) ) * (n + 1 - F(x)) = 0 for n > 0.
%C It is remarkable that this sequence should consist entirely of integers.
%e O.g.f.: A(x) = 1 + 3*x + 26*x^2 + 429*x^3 + 11140*x^4 + 413575*x^5 + 20732442*x^6 + 1349324599*x^7 + 110687183288*x^8 + 11178507440925*x^9 + ...
%e ILLUSTRATION OF DEFINITION.
%e The table of coefficients of x^k/k! in exp( n * x*A(x) ) * (n + 3 - A(x)) begins:
%e n=0: [2, -3, -52, -2574, -267360, -49629000, -14927358240, ...];
%e n=1: [3, 0, -37, -2268, -248745, -47203552, -14374490745, ...];
%e n=2: [4, 5, 0, -1462, -202352, -41536632, -13142258240, ...];
%e n=3: [5, 12, 65, 0, -121911, -32155140, -11164894659, ...];
%e n=4: [6, 21, 164, 2298, 0, -18516616, -8369990496, ...];
%e n=5: [7, 32, 303, 5636, 172075, 0, -4677722165, ...];
%e n=6: [8, 45, 488, 10242, 404400, 24104328, 0, ...];
%e n=7: [9, 60, 725, 16368, 708573, 54605228, 5760470145, 0, ...]; ...
%e in which the main diagonal is all zeros after the initial term, illustrating the property that [x^n] exp( n * x*A(x) ) * (n + 3 - A(x)) = 0 for n > 0.
%e RELATED SERIES.
%e exp(x*A(x)) = 1 + x + 7*x^2/2! + 175*x^3/3! + 11065*x^4/4! + 1399801*x^5/5! + 307183471*x^6/6! + 106838020087*x^7/7! + 55316481920785*x^8/8! + ...
%o (PARI) {a(n) = my(A=[1],m); for(i=1,n, A=concat(A,0); m=#A; A[m] = Vec( exp( (m-1)*x*(Ser(A)) ) * ((m-1) + 2 - Ser(A)) )[m] );A[n+1]}
%o for(n=0,20,print1(a(n),", "))
%Y Cf. A305110, A305111, A305112, A305114.
%K nonn
%O 0,2
%A _Paul D. Hanna_, May 26 2018
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