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%I #18 Jul 21 2024 00:10:39
%S 1,2,2,8,6,3,52,36,12,4,480,324,96,20,5,5816,3888,1104,200,30,6,87936,
%T 58536,16320,2800,360,42,7,1601728,1064016,294048,49200,5940,588,56,8,
%U 34251520,22728384,6252288,1032800,120960,11172,896,72,9,843099616
%N Triangular matrix T, read by rows, that satisfies: T^2 + T = SHIFTUP(T), also T^(n+1) + T^n = SHIFTUP(T^n - D*T^(n-1)) for all n, where D is a diagonal matrix with diagonal(D) = diagonal(T) = {1,2,3,...}.
%C Leftmost column is A103239. The operation SHIFTUP(T) shifts each column of T up 1 row, dropping the elements that occupied the diagonal of T.
%F G.f. for column k: T(k, k) = k+1 = Sum_{n>=k} T(n, k)*x^(n-k)/(1-x)^(n-k) * Product_{j=0..n-k} (1-(j+k+2)*x). Diagonalization: T = P*D*P^-1 where P(r, c) = A103249(r, c)/(r-c)! = (-1)^(r-c)*(c^2+c)^(r-c)/(r-c)! for r>=c>=1 and [P^-1](r, c) = A103244(r, c)/(r-c)! and D is a diagonal matrix = {1, 2, 3, ...}.
%e Rows of T begin:
%e [1],
%e [2,2],
%e [8,6,3],
%e [52,36,12,4],
%e [480,324,96,20,5],
%e [5816,3888,1104,200,30,6],
%e [87936,58536,16320,2800,360,42,7],
%e [1601728,1064016,294048,49200,5940,588,56,8],...
%e Rows of T^2 begin:
%e [1],
%e [6,4],
%e [44,30,9],
%e [428,288,84,16],
%e [5336,3564,1008,180,25],...
%e Then T^2 + T = SHIFTUP(T):
%e [2],
%e [8,6],
%e [52,36,12],
%e [480,324,96,20],
%e [5816,3888,1104,200,30],...
%e G.f. for column 0: 1 = (1-2x) + 2*x/(1-x)*(1-2x)(1-3x) + 8*x^2/(1-x)^2*(1-2x)(1-3x)(1-4x) + 52*x^3/(1-x)^3*(1-2x)(1-3x)(1-4x)(1-5x) + ... + T(n,0)*x^n/(1-x)^n*(1-2x)(1-3x)*..*(1-(n+2)x) + ...
%e G.f. for column 1: 2 = 2*(1-3x) + 6*x/(1-x)*(1-3x)(1-4x) + 36*x^2/(1-x)^2*(1-3x)(1-4x)(1-5x) + 324*x^3/(1-x)^3*(1-3x)(1-4x)(1-5x)(1-6x) + ... + T(n,1)*x^(n-1)/(1-x)^(n-1)*(1-3x)(1-4x)*..*(1-(n+2)x) + ...
%o (PARI) /* Using Matrix Diagonalization Formula: */ T(n,k)=my(P,D);D=matrix(n+1,n+1,r,c,if(r==c,r)); P=matrix(n+1,n+1,r,c,if(r>=c,(-1)^(r-c)*(c^2+c)^(r-c)/(r-c)!)); return(if(n<k||k<0,0,(P*D*P^-1)[n+1,k+1]))
%o (PARI) /* Using Generating Function for Columns: */ T(n,k)=if(n<k,0,if(n==k,k+1,polcoeff( k+1-sum(i=k,n-1,T(i,k)*x^(i-k)/(1-x)^(i-k)* prod(j=0,i-k,1-(j+k+2)*x+x*O(x^n))),n-k)))
%Y Cf. A103239, A103244, A103249, A103236, A103237.
%K nonn,tabl
%O 0,2
%A _Paul D. Hanna_, Jan 31 2005
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