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A103238
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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,...}.
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2
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1, 2, 2, 8, 6, 3, 52, 36, 12, 4, 480, 324, 96, 20, 5, 5816, 3888, 1104, 200, 30, 6, 87936, 58536, 16320, 2800, 360, 42, 7, 1601728, 1064016, 294048, 49200, 5940, 588, 56, 8, 34251520, 22728384, 6252288, 1032800, 120960, 11172, 896, 72, 9, 843099616
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OFFSET
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0,2
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COMMENTS
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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.
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LINKS
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FORMULA
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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, ...}.
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EXAMPLE
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Rows of T begin:
[1],
[2,2],
[8,6,3],
[52,36,12,4],
[480,324,96,20,5],
[5816,3888,1104,200,30,6],
[87936,58536,16320,2800,360,42,7],
[1601728,1064016,294048,49200,5940,588,56,8],...
Rows of T^2 begin:
[1],
[6,4],
[44,30,9],
[428,288,84,16],
[5336,3564,1008,180,25],...
Then T^2 + T = SHIFTUP(T):
[2],
[8,6],
[52,36,12],
[480,324,96,20],
[5816,3888,1104,200,30],...
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) + ...
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) + ...
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PROG
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(PARI) /* Using Matrix Diagonalization Formula: */ T(n, k)=local(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]))
(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)))
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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