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%I #8 Jan 04 2019 18:32:58
%S 1,1,1,1,2,1,1,3,6,1,1,4,15,23,1,1,5,28,102,104,1,1,6,45,274,861,537,
%T 1,1,7,66,575,3400,8598,3100,1,1,8,91,1041,9425,50734,98547,19693,1,1,
%U 9,120,1708,21216,187455,880312,1270160,136064,1,1,10,153,2612,41629
%N Rectangular array of coefficients in successive iterations of x*exp(x), as read by antidiagonals.
%C Triangle A174485 forms a matrix that transforms a diagonal into an adjacent diagonal in this array.
%F T(n,k) = [x^k/(k-1)! ] G_{n}(x) where G_{n}(x) = G_{n-1}(x*exp(x)) with G_0(x)=x, for n>=1, k>=1.
%e Form an array of coefficients in the iterations of x*exp(x), which begin:
%e n=1: [1, 1, 1/2!, 1/3!, 1/4!, 1/5!, 1/6!, ...];
%e n=2: [1, 2, 6/2!, 23/3!, 104/4!, 537/5!, 3100/6!, ...];
%e n=3: [1, 3, 15/2!, 102/3!, 861/4!, 8598/5!, 98547/6!, ...];
%e n=4: [1, 4, 28/2!, 274/3!, 3400/4!, 50734/5!, 880312/6!, ...];
%e n=5: [1, 5, 45/2!, 575/3!, 9425/4!, 187455/5!, 4367245/6!, ...];
%e n=6: [1, 6, 66/2!, 1041/3!, 21216/4!, 527631/5!, 15441636/6!, ...];
%e n=7: [1, 7, 91/2!, 1708/3!, 41629/4!, 1242892/5!, 43806175/6!, ...];
%e n=8: [1, 8, 120/2!, 2612/3!, 74096/4!, 2582028/5!, 106459312/6!, ...];
%e n=9: [1, 9, 153/2!, 3789/3!, 122625/4!, 4885389/5!, 230689017/6!, ...];
%e n=10:[1, 10, 190/2!, 5275/3!, 191800/4!, 8599285/5!, 457584940/6!,...];
%e ...
%e This array begins with the above unreduced numerators for n >= 1, k >= 1.
%o (PARI) {T(n, k)=local(F=x, xEx=x*exp(x+x*O(x^(k+1)))); for(i=1,n,F=subst(F, x, xEx));(k-1)!*polcoeff(F, k)}
%Y Cf. A174485, diagonals: A174481, A174482, A174483, A174484.
%Y Cf. rows: A080108, A174493, A174494, A174495.
%K nonn,tabl
%O 1,5
%A _Paul D. Hanna_, Apr 17 2010
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