A123305 Triangle T, read by rows, where column k of T equals (k+1)*(column k of T^2) when shifted to have an initial '1'; i.e., T(n,k) = (k+1)*[T^2](n-1,k) for n>k>=0, with T(n,n)=1 for n>=0.

1, 1, 1, 2, 2, 1, 6, 8, 3, 1, 26, 44, 18, 4, 1, 156, 312, 144, 32, 5, 1, 1234, 2776, 1422, 336, 50, 6, 1, 12340, 30312, 16848, 4256, 650, 72, 7, 1, 150994, 397728, 235458, 63072, 10050, 1116, 98, 8, 1, 2204112, 6151768, 3827628, 1076128, 178900, 20376, 1764

Offset: 0

Author

Paul D. Hanna, Sep 24 2006

Keywords

• nonn
• tabl

Examples

			Triangle begins:
  1;
  1, 1;
  2, 2, 1;
  6, 8, 3, 1;
  26, 44, 18, 4, 1;
  156, 312, 144, 32, 5, 1;
  1234, 2776, 1422, 336, 50, 6, 1;
  12340, 30312, 16848, 4256, 650, 72, 7, 1;
  150994, 397728, 235458, 63072, 10050, 1116, 98, 8, 1;
  ...
Matrix square starts:
  1;
  2, 1;
  6, 4, 1;
  26, 22, 6, 1;
  156, 156, 48, 8, 1;
  1234, 1388, 474, 84, 10, 1;
  12340, 15156, 5616, 1064, 130, 12, 1;
  ...
Note that (column k of T shifted) = (k+1)*(column k of T^2):
  k=1: [2,8,44,312,2776,...] = 2*[1,4,22,156,1388,...];
  k=2: [3,18,144,1422,16848,....] = 3*[1,6,48,474,5616,...].
		

Crossrefs

Columns: A123306, A123307, A123308, A123309.

Cf. A123310 (row sums), A123311 (central terms).

Programs

PARI

{T(n, k)=if(n<0||k>n,0,if(n==k,1,(k+1)*sum(j=0, n-1, T(n-1, j)*T(j, k)); ))}

Formula

T(n,k) = (k+1)*Sum_{j=0..n-1} T(n-1,j)*T(j,k) for n>0, with T(n,n)=1 n>=0.