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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.

%I A103244 #19 Aug 09 2018 05:03:54
%S A103244 1,2,1,20,6,1,512,108,12,1,25392,4104,336,20,1,2093472,273456,17568,
%T A103244 800,30,1,260555392,28515456,1500288,54800,1620,42,1,45819233280,
%U A103244 4311418752,191549952,5808000,140400,2940,56,1,10849051434240,894918533760,34352605440,887256000,18033840,313992,4928,72,1
%N 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.
%C A103244 Define triangular matrix P by P(n,k) = (-k^2-k)^(n-k)/(n-k)!, then M = P*D*P^-1 = A103238 satisfies: M^2 + M = SHIFTUP(M) where D is the diagonal matrix consisting of {1,2,3,...}. The operation SHIFTUP(M) shifts each column of M up 1 row.
%C A103244 First column is A103353.
%F A103244 For n > k >= 1: 0 = Sum_{m=k..n} C(n-k, m-k)*(-m^2-m)^(n-m)*T(m, k).
%F A103244 For n > k >= 1: 0 = Sum_{j=k..n} C(n-k, j-k)*(-k^2-k)^(j-k)*T(n, j).
%e A103244 This triangle begins:
%e A103244 1;
%e A103244 2, 1;
%e A103244 20, 6, 1;
%e A103244 512, 108, 12, 1;
%e A103244 25392, 4104, 336, 20, 1;
%e A103244 2093472, 273456, 17568, 800, 30, 1;
%e A103244 260555392, 28515456, 1500288, 54800, 1620, 42, 1;
%e A103244 45819233280, 4311418752, 191549952, 5808000, 140400, 2940, 56, 1;
%e A103244 10849051434240, 894918533760, 34352605440, 887256000, 18033840, 313992, 4928, 72, 1; ...
%e A103244 Rows of unreduced fractions T(n,k)/(n-k)! begin:
%e A103244 [1/0!],
%e A103244 [2/1!, 1/0!],
%e A103244 [20/2!, 6/1!, 1/0!],
%e A103244 [512/3!, 108/2!, 12/1!, 1/0!],
%e A103244 [25392/4!, 4104/3!, 336/2!, 20/1!, 1/0!],
%e A103244 [2093472/5!, 273456/4!, 17568/3!, 800/2!, 30/1!, 1/0!],...
%e A103244 forming the inverse of matrix P where P(n,k) = (-1)^(n-k)*(k^2+k)^(n-k)/(n-k)!:
%e A103244 [1/0!],
%e A103244 [ -2/1!, 1/0!],
%e A103244 [4/2!, -6/1!, 1/0!],
%e A103244 [ -8/3!, 36/2!, -12/1!, 1/0!],
%e A103244 [16/4!, -216/3!, 144/2!, -20/1!, 1/0!], ...
%t A103244 nmax = 9;
%t A103244 P = Table[If[n >= k, (-k^2-k)^(n-k)/(n-k)!, 0], {n, 1, nmax}, {k, 1, nmax}] // Inverse;
%t A103244 T[n_, k_] := If[n < k || k < 1, 0, P[[n, k]]*(n - k)!];
%t A103244 Table[T[n, k], {n, 1, nmax}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Aug 09 2018, from PARI *)
%o A103244 (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<k||k<1,0,(P^-1)[n,k]*(n-k)!))}
%o A103244 for(n=1,10,for(k=1,n,print1(T(n,k),", "));print(""))
%Y A103244 Cf. A261642, A103238, A103353.
%K A103244 nonn,tabl,frac
%O A103244 1,2
%A A103244 _Paul D. Hanna_, Feb 02 2005