A103247 -id:A103247 - cp's OEIS frontend

A107056 Matrix inverse of A103247, so that T(n,k) = C(n,k)*A010842(n-k), read by rows.

1, 3, 1, 10, 6, 1, 38, 30, 9, 1, 168, 152, 60, 12, 1, 872, 840, 380, 100, 15, 1, 5296, 5232, 2520, 760, 150, 18, 1, 37200, 37072, 18312, 5880, 1330, 210, 21, 1, 297856, 297600, 148288, 48832, 11760, 2128, 280, 24, 1, 2681216, 2680704, 1339200, 444864, 109872

Offset: 0

Paul D. Hanna, May 19 2005

A103247(n,k) is the coefficient of x^k in the monic characteristic polynomial of the n X n matrix with 3's on the diagonal and 1's elsewhere.

			Triangle T begins:
1;
3,1;
10,6,1;
38,30,9,1;
168,152,60,12,1;
872,840,380,100,15,1;
5296,5232,2520,760,150,18,1; ...
where T(n,k) = A010842(n-k)*binomial(n,k).
Matrix logarithm L begins:
0;
-3,0;
-1,-6,0;
-2,-3,-9,0;
-6,-8,-6,-12,0;
-24,-30,-20,-10,-15,0; ...
where L(n,k) = L(n,0)*binomial(n,k),
with L(n,0)=-(n-1)! for n>1, L(1,0)=-3, L(0,0)=0.

Cf. A103247, A010842.

T(n,k)=n!/k!*sum(j=0,n-k,2^(n-k-j)/(n-k-j)!)

T(n, k) = n!/k!*Sum_{j=0..n-k} 2^(n-k-j)/(n-k-j)!.

A103236 Triangular matrix T, read by rows, that satisfies: T^2 + 2T = SHIFTUP(T), also T^(n+1) + 2T^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,...}.

Original entry on oeis.org

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

Offset: 0

Paul D. Hanna, Jan 31 2005

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.

			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) + ...

Cf. A082163, A103247, A103242.

PARI
```
{T(n,k)=if(n
				
```

G.f. for column k: T(k, k) = k+1 = Sum_{n>=k} T(n, k)*x^(n-k)/(1-2*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, ...}.

A103242 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)^2)^(n-k)/(n-k)! for n >= k >= 1.

Original entry on oeis.org

1, 3, 1, 39, 8, 1, 1206, 176, 15, 1, 69189, 7784, 495, 24, 1, 6416568, 585408, 29430, 1104, 35, 1, 881032059, 67481928, 2791125, 84600, 2135, 48, 1, 168514815360, 11111547520, 389244600, 9841728, 204470, 3744, 63, 1, 42934911510249

Offset: 1

Paul D. Hanna, Feb 02 2005

Define a triangular matrix P where P(n,k) = (-k^2-2*k)^(n-k)/(n-k)!; then M = P*D*P^-1 = A103236 satisfies M^2 + 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. Essentially equal to square array A082171 as a triangular matrix. The first column is A082163 (enumerates acyclic automata with 2 inputs).

			Rows of unreduced fractions T(n,k)/(n-k)! begin:
  [1/0!],
  [3/1!, 1/0!],
  [39/2!, 8/1!, 1/0!],
  [1206/3!, 176/2!, 15/1!, 1/0!],
  [69189/4!, 7784/3!, 495/2!, 24/1!, 1/0!],
  [6416568/5!, 585408/4!, 29430/3!, 1104/2!, 35/1!, 1/0!], ...
forming the inverse of matrix P where P(n,k) = A103247(n,k)/(n-k)!:
  [1/0!],
  [ -3/1!, 1/0!],
  [9/2!, -8/1!, 1/0!],
  [ -27/3!, 64/2!, -15/1!, 1/0!],
  [81/4!, -512/3!, 225/2!, -24/1!, 1/0!],
  [ -243/5!, 4096/4!, -3375/3!, 576/2!, -35/1!, 1/0!], ...

Cf. A103247, A103236, A082171, A082163.

{T(n,k)=my(P);if(n>=k&k>=1, P=matrix(n,n,r,c,if(r>=c,(1-(c+1)^2)^(r-c)/(r-c)!))); return(if(n

For n > k >= 1: 0 = Sum_{m=k..n} C(n-k, m-k)*(1-(m+1)^2)^(n-m)*T(m, k).

For n > k >= 1: 0 = Sum_{j=k..n} C(n-k, j-k)*(1-(k+1)^2)^(j-k)*T(n, j).

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).

A103407 Triangle of absolute values of the coefficients (in descending powers) of the characteristic polynomials of n X n matrices with 3's on the main diagonal and 1's elsewhere.

Original entry on oeis.org

1, 1, 3, 1, 6, 8, 1, 9, 24, 20, 1, 12, 48, 80, 48, 1, 15, 80, 200, 240, 112, 1, 18, 120, 400, 720, 672, 256, 1, 21, 168, 700, 1680, 2352, 1792, 576, 1, 24, 224, 1120, 3360, 6272, 7168, 4608, 1280, 1, 27, 288, 1680, 6048, 14112, 21504, 20736, 11520, 2816, 1, 30

Offset: 0

Gary W. Adamson, Feb 04 2005

			3rd row (1, 9, 24, 20) with alternating signs = characteristic polynomial 3 X 3 matrix [3 1 1 / 1 3 1 / 1 1 3], x^3 - 9x^2 + 24x - 20.

Row sums are A006234: 1, 4, 15, 54, 189... Rightmost terms in each row = A001792: 1, 3, 8, 20, 48, 112, 256...(row sums of A103406, the analogous triangle with all 2's in the generating matrix.)

See A103247 for another version.

Extended and edited by John W. Layman, Mar 17 2005

cp's OEIS Frontend

A107056 Matrix inverse of A103247, so that T(n,k) = C(n,k)*A010842(n-k), read by rows.

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Formula

A103236 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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Formula

A103242 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)^2)^(n-k)/(n-k)! for n >= k >= 1.

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PARI

Formula

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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Author

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Examples

Crossrefs

Programs

PARI

Formula

A103407 Triangle of absolute values of the coefficients (in descending powers) of the characteristic polynomials of n X n matrices with 3's on the main diagonal and 1's elsewhere.

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Extensions

A103236 Triangular matrix T, read by rows, that satisfies: T^2 + 2T = SHIFTUP(T), also T^(n+1) + 2T^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,...}.