A101515 Symmetric square array, read by antidiagonals, such that the inverse binomial transform of row n forms the sequence: {C(n,k)*A101514(k), 0<=k<=n}, where A101514 equals the main diagonal shift right.

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 7, 4, 1, 1, 5, 13, 13, 5, 1, 1, 6, 21, 35, 21, 6, 1, 1, 7, 31, 77, 77, 31, 7, 1, 1, 8, 43, 146, 236, 146, 43, 8, 1, 1, 9, 57, 249, 596, 596, 249, 57, 9, 1, 1, 10, 73, 393, 1290, 2037, 1290, 393, 73, 10, 1, 1, 11, 91, 585, 2486, 5772, 5772, 2486, 585

Offset: 0

Author

Paul D. Hanna, Dec 06 2004

Keywords

• nonn
• tabl

Comments

The main diagonal equals A101514 shift one place left. The antidiagonal sums form A101516.

Examples

			Rows begin:
[_1,1,1,1,1,1,1,1,1,...],
[1,_2,3,4,5,6,7,8,9,...],
[1,3,_7,13,21,31,43,57,73,...],
[1,4,13,_35,77,146,249,393,585,...],
[1,5,21,77,_236,596,1290,2486,4387,...],
[1,6,31,146,596,_2037,5772,13987,29987,...],
[1,7,43,249,1290,5772,_21695,67943,181811,...],
[1,8,57,393,2486,13987,67943,_277966,951051,...],
[1,9,73,585,4387,29987,181811,951051,_4198635,...],...
The inverse binomial transform of the rows of this array are generated
from the products of the main diagonal with rows of Pascal's triangle:
BINOMIAL[1*1] = [_1,1,1,1,1,1,1,1,1,...],
BINOMIAL[1*1,1*1] = [1,_2,3,4,5,6,7,8,9,...],
BINOMIAL[1*1,1*2,2*1] = [1,3,_7,13,21,31,43,57,73,...],
BINOMIAL[1*1,1*3,2*3,7*1] = [1,4,13,_35,77,146,249,393,...],
BINOMIAL[1*1,1*4,2*6,7*4,35*1] = [1,5,21,77,_236,596,1290,...],
BINOMIAL[1*1,1*5,2*10,7*10,35*5,236*1] = [1,6,31,146,596,_2037,...],...
		

Crossrefs

Cf. A101514, A101516.

Programs

PARI

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