A103237 -id:A103237 - cp's OEIS frontend

A103238 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,...}.

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

Offset: 0

Paul D. Hanna, Jan 31 2005

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.

			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) + ...

Cf. A103239, A103244, A103249, A103236, A103237.

/* 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

/* Using Generating Function for Columns: */ T(n,k)=if(n

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, ...}.

A103243 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) = (1-(k+1)^3)^(n-k)/(n-k)! for n >= k >= 1.

Original entry on oeis.org

1, 7, 1, 315, 26, 1, 45682, 2600, 63, 1, 15646589, 675194, 11655, 124, 1, 10567689552, 366349152, 4861458, 37944, 215, 1, 12503979423607, 361884843866, 3882676581, 23641468, 100835, 342, 1, 23841011541867520, 591934698991168, 5318920238688

Offset: 1

Paul D. Hanna, Feb 02 2005

Define triangular matrix P by P(n,k) = (-k^3-3k^2-3k)^(n-k)/(n-k)!, then M = P*D*P^-1 = A103237 satisfies: M^3 + 3M^2 + 3M = 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. Essentially equal to square array A082172 as a triangular matrix. The first column is A082160 (quasi-acyclic automata with 3 inputs).

			Rows of unreduced fractions T(n,k)/(n-k)! begin:
  [1/0! ],
  [7/1!, 1/0! ],
  [315/2!, 26/1!, 1/0! ],
  [45682/3!, 2600/2!, 63/1!, 1/0! ],
  [15646589/4!, 675194/3!, 11655/2!, 124/1!, 1/0! ],
  [10567689552/5!, 366349152/4!, 4861458/3!, 37944/2!, 215/1!, 1/0! ], ...
forming the inverse of matrix P where P(n,k) = A103247(n,k)/(n-k)!:
  [1/0! ],
  [ -7/1!, 1/0! ],
  [49/2!, -26/1!, 1/0! ],
  [ -343/3!, 676/2!, -63/1!, 1/0! ],
  [2401/4!, -17576/3!, 3969/2!, -124/1!, 1/0! ],
  [ -16807/5!, 456976/4!, -250047/3!, 15376/2!, -215/1!, 1/0! ], ...

Cf. A103248, A103237, A082172, A082160.

{T(n,k)=my(P);if(n>=k&k>=1, P=matrix(n,n,r,c,if(r>=c,(1-(c+1)^3)^(r-c)/(r-c)!))); return(if(n

For n > k >= 1: 0 = Sum_{m=k..n} C(n-k, m-k)*(1-(m+1)^3)^(n-m)*T(m, k). For n > k >= 1: 0 = Sum_{j=k..n} C(n-k, j-k)*(1-(k+1)^3)^(j-k)*T(n, j).

cp's OEIS Frontend

A103238 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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A103243 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) = (1-(k+1)^3)^(n-k)/(n-k)! for n >= k >= 1.

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