A103244 Unreduced numerators of the elements T(n,k)/(n-k)!, read by rows, of the triangular matrix P^-1, which is the inverse of the matrix defined by P(n,k) = (-k^2-k)^(n-k)/(n-k)! for n >= k >= 1.
1, 2, 1, 20, 6, 1, 512, 108, 12, 1, 25392, 4104, 336, 20, 1, 2093472, 273456, 17568, 800, 30, 1, 260555392, 28515456, 1500288, 54800, 1620, 42, 1, 45819233280, 4311418752, 191549952, 5808000, 140400, 2940, 56, 1, 10849051434240, 894918533760, 34352605440, 887256000, 18033840, 313992, 4928, 72, 1
Offset: 1
Examples
This triangle begins: 1; 2, 1; 20, 6, 1; 512, 108, 12, 1; 25392, 4104, 336, 20, 1; 2093472, 273456, 17568, 800, 30, 1; 260555392, 28515456, 1500288, 54800, 1620, 42, 1; 45819233280, 4311418752, 191549952, 5808000, 140400, 2940, 56, 1; 10849051434240, 894918533760, 34352605440, 887256000, 18033840, 313992, 4928, 72, 1; ... Rows of unreduced fractions T(n,k)/(n-k)! begin: [1/0!], [2/1!, 1/0!], [20/2!, 6/1!, 1/0!], [512/3!, 108/2!, 12/1!, 1/0!], [25392/4!, 4104/3!, 336/2!, 20/1!, 1/0!], [2093472/5!, 273456/4!, 17568/3!, 800/2!, 30/1!, 1/0!],... forming the inverse of matrix P where P(n,k) = (-1)^(n-k)*(k^2+k)^(n-k)/(n-k)!: [1/0!], [ -2/1!, 1/0!], [4/2!, -6/1!, 1/0!], [ -8/3!, 36/2!, -12/1!, 1/0!], [16/4!, -216/3!, 144/2!, -20/1!, 1/0!], ...
Programs
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Mathematica
nmax = 9; P = Table[If[n >= k, (-k^2-k)^(n-k)/(n-k)!, 0], {n, 1, nmax}, {k, 1, nmax}] // Inverse; T[n_, k_] := If[n < k || k < 1, 0, P[[n, k]]*(n - k)!]; Table[T[n, k], {n, 1, nmax}, {k, 1, n}] // Flatten (* Jean-François Alcover, Aug 09 2018, from PARI *) -
PARI
{T(n,k)=local(P);if(n>=k&k>=1, P=matrix(n,n,r,c,if(r>=c,(-c^2-c)^(r-c)/(r-c)!))); return(if(n
Formula
For n > k >= 1: 0 = Sum_{m=k..n} C(n-k, m-k)*(-m^2-m)^(n-m)*T(m, k).
For n > k >= 1: 0 = Sum_{j=k..n} C(n-k, j-k)*(-k^2-k)^(j-k)*T(n, j).
Comments