A102321 -id:A102321 - cp's OEIS frontend

A102320 Triangular matrix, read by rows, that satisfies: T(n,k) = [T^2](n-1,k) when n>k>=0, with T(n,n) = (2*n+1).

1, 1, 3, 4, 9, 5, 33, 72, 25, 7, 436, 945, 300, 49, 9, 8122, 17568, 5425, 784, 81, 11, 197920, 427770, 130700, 18081, 1620, 121, 13, 6007205, 12979080, 3947050, 535864, 45441, 2904, 169, 15, 219413116, 473981445, 143812400, 19348042, 1599588, 95953

Offset: 0

Paul D. Hanna, Jan 05 2005

Column 0 forms A102321. Column 1 forms A102322. The contribution of each term along the main diagonal to column 0 is given by triangle of coefficients A102323.

			Rows of T begin:
[1],
[1,3],
[4,9,5],
[33,72,25,7],
[436,945,300,49,9],
[8122,17568,5425,784,81,11],
[197920,427770,130700,18081,1620,121,13],
[6007205,12979080,3947050,535864,45441,2904,169,15],...
Matrix square T^2 equals T excluding the main diagonal:
[1],
[4,9],
[33,72,25],
[436,945,300,49],
[8122,17568,5425,784,81],...

Cf. A102086, A102321, A102322, A102323.

{T(n,k)=local(A=Mat(1),B); for(m=2,n+1,B=matrix(m,m);for(i=1,m, for(j=1,i, if(j==i,B[i,j]=2*j-1,if(j==1,B[i,j]=(A^2)[i-1,1], B[i,j]=(A^2)[i-1,j]));));A=B);return(A[n+1,k+1])}

A102323 Triangle, read by rows, where T(n,k) = T(n,k-1) + (2k+1)T(n-1,k) for n>k>0, T(n,0)=1 and T(n,n) = T(n,n-1) for n>=0.

Original entry on oeis.org

1, 1, 1, 1, 4, 4, 1, 13, 33, 33, 1, 40, 205, 436, 436, 1, 121, 1146, 4198, 8122, 8122, 1, 364, 6094, 35480, 108578, 197920, 197920, 1, 1093, 31563, 279923, 1257125, 3434245, 6007205, 6007205, 1, 3280, 161095, 2120556, 13434681, 51211376

Offset: 0

Paul D. Hanna, Jan 05 2005

Main diagonal is A102321, which is column 0 of triangle A102320.

			T(5,2) = 1146 = 1*1 + 3*40 + 5*205 = 1*T(4,0) + 3*T(4,1) + 5*T(4,2).
T(5,2) = 1146 = 121 + 5*205 = T(5,1) + (2*2+1)*T(4,2).
T(5,3) = 4198 = 1146 + 7*436 = T(5,2) + (2*3+1)*T(4,3).
Rows begin:
[1],
[1,1],
[1,4,4],
[1,13,33,33],
[1,40,205,436,436],
[1,121,1146,4198,8122,8122],
[1,364,6094,35480,108578,197920,197920],
[1,1093,31563,279923,1257125,3434245,6007205,6007205],...

Cf. A102316, A102321, A102320.

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

cp's OEIS Frontend

A102320 Triangular matrix, read by rows, that satisfies: T(n,k) = [T^2](n-1,k) when n>k>=0, with T(n,n) = (2*n+1).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

A102323 Triangle, read by rows, where T(n,k) = T(n,k-1) + (2k+1)T(n-1,k) for n>k>0, T(n,0)=1 and T(n,n) = T(n,n-1) for n>=0.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

cp's OEIS Frontend

A102320 Triangular matrix, read by rows, that satisfies: T(n,k) = [T^2](n-1,k) when n>k>=0, with T(n,n) = (2*n+1).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

A102323 Triangle, read by rows, where T(n,k) = T(n,k-1) + (2*k+1)*T(n-1,k) for n>k>0, T(n,0)=1 and T(n,n) = T(n,n-1) for n>=0.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

A102323 Triangle, read by rows, where T(n,k) = T(n,k-1) + (2k+1)T(n-1,k) for n>k>0, T(n,0)=1 and T(n,n) = T(n,n-1) for n>=0.