A174485 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).
1, 1, 1, 5, 2, 1, 70, 16, 3, 1, 1973, 308, 33, 4, 1, 94216, 11048, 810, 56, 5, 1, 6851197, 639972, 35325, 1672, 85, 6, 1, 706335064, 54671188, 2408568, 85904, 2990, 120, 7, 1, 98105431657, 6471586298, 236624733, 6741544, 176885, 4860, 161, 8, 1
Offset: 0
Examples
Triangle T begins: 1; 1,1; 5,2,1; 70,16,3,1; 1973,308,33,4,1; 94216,11048,810,56,5,1; 6851197,639972,35325,1672,85,6,1; 706335064,54671188,2408568,85904,2990,120,7,1; 98105431657,6471586298,236624733,6741544,176885,4860,161,8,1; 17669939141440,1014487323984,31654735416,749040472,15706200,325368,7378,208,9,1; ... Form a table of coefficients in iterations of x*exp(x), like so: n=0: [1, 0, 0, 0, 0, 0, 0, ...]; n=1: [1, 1, 1/2!, 1/3!, 1/4!, 1/5!, 1/6!, ...]; n=2: [1, 2, 6/2!, 23/3!, 104/4!, 537/5!, 3100/6!, ...]; n=3: [1, 3, 15/2!, 102/3!, 861/4!, 8598/5!, 98547/6!, ...]; n=4: [1, 4, 28/2!, 274/3!, 3400/4!, 50734/5!, 880312/6!, ...]; n=5: [1, 5, 45/2!, 575/3!, 9425/4!, 187455/5!, 4367245/6!, ...]; n=6: [1, 6, 66/2!, 1041/3!, 21216/4!, 527631/5!+ 15441636/6!, ...]; n=7: [1, 7, 91/2!, 1708/3!, 41629/4!, 1242892/5!, 43806175/6!, ...]; n=8: [1, 8, 120/2!, 2612/3!, 74096/4!, 2582028/5!, 106459312/6!, ...]; ... and form matrix D from this triangle T by: D(n,k) = T(n,k)/(n-k)!, then matrix D transforms diagonals in the above table as illustrated by: D * A174481 = A174482, D * A174482 = A174483, D * A174483 = A174484, where the diagonals begin: A174481: [1, 1, 6/2!, 102/3!, 3400/4!, 187455/5!, ...]; A174482: [1, 2, 15/2!, 274/3!, 9425/4!, 527631/5!, ...]; A174483: [1, 3, 28/2!, 575/3!, 21216/4!, 1242892/5!, ...]; A174484: [1, 4, 45/2!, 1041/3!, 41629/4!, 2582028/5!, ...].
Crossrefs
Programs
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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]}
Comments