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%I #10 Aug 31 2013 09:41:52
%S 1,2,-30,1008,-50760,3227400,-232071840,17196863040,-1246907208960,
%T 104187854836800,-13090506064574400,191142937964563200,
%U 646777849055450112000,339995571993104227200000,-227122463058126580264320000,-361611207685046931735771648000,110410149903015181792955526144000
%N E.g.f. A(x) satisfies the property that the coefficient of x^n in the (2*n)-th iterations of e.g.f. A(x), for n>=1, begins with [1,4] and continues with all zeros thereafter.
%e G.f.: A(x) = x + 2*x^2/2! - 30*x^3/3! + 1008*x^4/4! - 50760*x^5/5! + 3227400*x^6/6! - 232071840*x^7/7! + 17196863040*x^8/8! +...
%e Coefficients of x^k/k! in the even iterations of the g.f. A(x) begin:
%e n=2: [1, 4, -48, 1440, -65280, 3628800,-209986560, 9686476800, ...];
%e n=4: [1, 8, -48, 1152, -40320, 1336320, 11934720,-11065098240, ...];
%e n=6: [1, 12, 0, 288, -5760, -518400, 106444800,-11752151040, ...];
%e n=8: [1, 16, 96, 0, 3840, -691200, 67737600, -3261726720, ...];
%e n=10:[1, 20, 240, 1440, 0, -288000, 13870080, 1614735360, ...];
%e n=12:[1, 24, 432, 5760, 40320, 0, -10644480, 1403781120, ...];
%e n=14:[1, 28, 672, 14112, 228480, 2661120, 0, -1079285760, ...];
%e n=16:[1, 32, 960, 27648, 714240, 16128000, 279982080, 0, ...]; ...
%e where the main diagonal consists of all zeros for n>2.
%o (PARI) {ITERATE(n, F)=local(G=x); for(i=1, n, G=subst(G, x, F)); G}
%o {a(n)=local(A=[1, 2]); for(m=3, n, A=concat(A, 0); A[#A]=-(#A-1)!/2*Vec(ITERATE(2*(#A), sum(k=1, #A-1, A[k]*x^k/k!)+x*O(x^#A)))[#A]); A[n]}
%o for(n=1, 20, print1(a(n), ", "))
%Y Cf. A228508, A210723.
%K sign
%O 1,2
%A _Paul D. Hanna_, Aug 29 2013
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