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A103236
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Triangular matrix T, read by rows, that satisfies: T^2 + 2*T = SHIFTUP(T), also T^(n+1) + 2*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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4
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1, 3, 2, 15, 8, 3, 114, 56, 15, 4, 1191, 568, 135, 24, 5, 15993, 7536, 1710, 264, 35, 6, 263976, 123704, 27495, 4008, 455, 48, 7, 5189778, 2425320, 533565, 75696, 8050, 720, 63, 8, 118729335, 55403008, 12121920, 1695528, 174615, 14544, 1071, 80, 9
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OFFSET
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0,2
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COMMENTS
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Leftmost column is A082163 (enumerates acyclic automata with 2 inputs). 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)/(1-2*x)^(n-k) * Product_{j=0..n-k} (1-(j+k+3)*x). Diagonalization: T = P*D*P^-1 where P(r, c) = A103247(r, c)/(r-c)! = (-1)^(r-c)*(c^2+2*c)^(r-c)/(r-c)! for r>=c>=1 and [P^-1](r, c) = A103242(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],
[3,2],
[15,8,3],
[114,56,15,4],
[1191,568,135,24,5],
[15993,7536,1710,264,35,6],
[263976,123704,27495,4008,455,48,7],
[5189778,2425320,533565,75696,8050,720,63,8],...
Rows of T^2 begin:
[1],
[9,4],
[84,40,9],
[963,456,105,16],
[13611,6400,1440,216,25],...
Rows of T^2+2*T equals SHIFTUP(T):
[3],
[15,8],
[114,56,15],
[1191,568,135,24],
[15993,7536,1710,264,35],...
G.f. for column 0: 1 = (1-3x) + 3*x/(1-2x)*(1-3x)(1-4x) + 15*x^2/(1-2x)^2*(1-3x)(1-4x)(1-5x) + 114*x^3/(1-2x)^3*(1-3x)(1-4x)(1-5x)(1-6x) + ... + T(n,0)*x^n/(1-2*x)^n*(1-3x)(1-4x)*..*(1-(n+3)x) + ...
G.f. for column 1: 2 = 2*(1-4x) + 8*x/(1-2x)*(1-4x)(1-5x) + 56*x^2/(1-2x)^2*(1-4x)(1-5x)(1-6x) + 568*x^3/(1-2x)^3*(1-4x)(1-5x)(1-6x)(1-7x) + ... + T(n,1)*x^(n-1)/(1-2*x)^(n-1)*(1-4x)(1-5x)*..*(1-(n+3)x) + ...
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PROG
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(PARI) {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/(1-2*x)^(i-k)* prod(j=0, i-k, 1-(j+k+3)*x+x*O(x^n))), n)))}
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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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