A080419 Triangle of generalized Chebyshev coefficients.

1, 4, 1, 15, 7, 1, 54, 36, 10, 1, 189, 162, 66, 13, 1, 648, 675, 360, 105, 16, 1, 2187, 2673, 1755, 675, 153, 19, 1, 7290, 10206, 7938, 3780, 1134, 210, 22, 1, 24057, 37908, 34020, 19278, 7182, 1764, 276, 25, 1, 78732, 137781, 139968, 91854, 40824, 12474, 2592

Offset: 1

Author

Paul Barry, Feb 19 2003

Keywords

• easy
• nonn
• tabl

Comments

Second binomial transform of 'pruned' Pascal triangle Binomial(i+1,j+1), (i,j>=0).

Examples

			Rows are:
{1},
{4,1},
{15,7,1},
{54,36,10,1},
{189,162,66,13,1},
...
For example, 10 = 7+3*1, 66 = 36+3*10.
		

Crossrefs

Columns include A006234, A080420, A080421, A080422, A080423.

Programs

PARI

T(n, k) = if (k==1, (n+2)*3^(n-2), if (k==n, 1, if (k < n, T(n-1, k-1) + 3*T(n-1, k), 0)));
tabl(nn) = for (n=1, nn, for (k=1, n, print1(T(n, k), ", ")); print); \\ Michel Marcus, Apr 15 2018

Formula

T(n,1) = A006234(n+2), T(n,n) = 1, T(n,k) = T(n-1,k-1) + 3*T(n-1,k), T(n,k)=0 for k>n. - corrected by Michel Marcus, Apr 15 2018

As a square array, T1(n, k)= (n+3k)3^n Product{j=1..(k-1), n+j}/(3k(k-1)!) (k>=1, n>=0).