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%I #2 Mar 30 2012 18:37:21
%S 1,1,1,5,2,1,70,16,3,1,1973,308,33,4,1,94216,11048,810,56,5,1,6851197,
%T 639972,35325,1672,85,6,1,706335064,54671188,2408568,85904,2990,120,7,
%U 1,98105431657,6471586298,236624733,6741544,176885,4860,161,8,1
%N Triangle of numerators T(n,k) in the matrix {T(n,k)/(n-k)!,n>=k>=0} that transforms diagonals of the array (A174480) of coefficients in successive iterations of x*exp(x).
%e Triangle T begins:
%e 1;
%e 1,1;
%e 5,2,1;
%e 70,16,3,1;
%e 1973,308,33,4,1;
%e 94216,11048,810,56,5,1;
%e 6851197,639972,35325,1672,85,6,1;
%e 706335064,54671188,2408568,85904,2990,120,7,1;
%e 98105431657,6471586298,236624733,6741544,176885,4860,161,8,1;
%e 17669939141440,1014487323984,31654735416,749040472,15706200,325368,7378,208,9,1;
%e ...
%e Form a table of coefficients in iterations of x*exp(x), like so:
%e n=0: [1, 0, 0, 0, 0, 0, 0, ...];
%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 ...
%e and form matrix D from this triangle T by: D(n,k) = T(n,k)/(n-k)!,
%e then matrix D transforms diagonals in the above table as illustrated by:
%e D * A174481 = A174482, D * A174482 = A174483, D * A174483 = A174484,
%e where the diagonals begin:
%e A174481: [1, 1, 6/2!, 102/3!, 3400/4!, 187455/5!, ...];
%e A174482: [1, 2, 15/2!, 274/3!, 9425/4!, 527631/5!, ...];
%e A174483: [1, 3, 28/2!, 575/3!, 21216/4!, 1242892/5!, ...];
%e A174484: [1, 4, 45/2!, 1041/3!, 41629/4!, 2582028/5!, ...].
%o (PARI) {T(n, k)=local(F=x, xEx=x*exp(x+x*O(x^(n+2))), M, N, P, m=max(n, k)); M=matrix(m+2, m+2, r, c, F=x; for(i=1, r+c-2, F=subst(F, x, xEx)); polcoeff(F, c)); N=matrix(m+1, m+1, r, c, M[r, c]); P=matrix(m+1, m+1, r, c, M[r+1, c]); (n-k)!*(P~*N~^-1)[n+1, k+1]}
%Y Cf. columns: A174486, A174487, A174488, A174489.
%Y Cf. A174480, A174481, A174482, A174483, A174484.
%K nonn,tabl
%O 0,4
%A _Paul D. Hanna_, Apr 18 2010
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