A127714 Triangle, read by rows of (n+1)*(n+2)/2 terms, generated by the following rule. Start with a single '1' in row n=0; from then on, obtain row n from row n-1 by inserting zeros in row n-1 at positions: {(j+1)*n - j*(j-1)/2 | j=0..n} and then take partial sums.

1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 1, 3, 5, 5, 8, 11, 11, 14, 14, 14, 1, 4, 9, 14, 14, 22, 33, 44, 44, 58, 72, 72, 86, 86, 86, 1, 5, 14, 28, 42, 42, 64, 97, 141, 185, 185, 243, 315, 387, 387, 473, 559, 559, 645, 645, 645, 1, 6, 20, 48, 90, 132, 132, 196, 293, 434, 619, 804, 804

Offset: 0

Author

Paul D. Hanna, Jan 24 2007

Keywords

• nonn
• tabl

Examples

			Triangle begins:
  1;
  1, 1, 1;
  1, 2, 2, 3, 3, 3;
  1, 3, 5, 5, 8, 11, 11, 14, 14, 14;
  1, 4, 9, 14, 14, 22, 33, 44, 44, 58, 72, 72, 86, 86, 86;
  1, 5, 14, 28, 42, 42, 64, 97, 141, 185, 185, 243, 315, 387, 387, 473, 559, 559, 645, 645, 645;
  1, 6, 20, 48, 90, 132, 132, 196, 293, 434, 619, 804, 804, 1047, 1362, 1749, 2136, 2136, 2609, 3168, 3727, 3727, 4372, 5017, 5017, 5662, 5662, 5662;
  ...
Obtain row n from row n-1 by inserting zeros in row n-1 at positions:
[n,2*n,3*n-1,4*n-3,5*n-6,6*n-10,...,(j+1)*n - j*(j-1)/2,... | j=0..n],
and then take partial sums; illustrated by the following examples.
Obtain row 3 from row 2 by inserting zeros at positions [3,6,8,9],
and then take partial sums:
[1, 2, 2, 0, 3, 3, 0, 3, 0, 0];
[1, 3, 5, 5, 8,11,11,14,14,14];
Obtain row 4 from row 3 by inserting zeros at positions [4,8,11,13,14],
and then take partial sums:
[1, 3, 5, _5, _0, _8, 11, 11, _0, 14, 14, _0, 14, _0, _0];
[1, 4, 9, 14, 14, 22, 33, 44, 44, 58, 72, 72, 86, 86, 86].
		

Crossrefs

Cf. A127715, A127716; A127632, A000108, A009766; variants: A127452, A127420.

Programs

PARI

{T(n,k)=local(t);if(n<0 || k<0 || k>(n+1)*(n+2)/2-1,0, t=(sqrtint((2*n+3)^2-8*(k+1))-1)\2; if(k==0,1,if(issquare((2*n+3)^2-8*(k+1)),T(n,k-1),T(n,k-1)+T(n-1,k-n+t))))} {/* for(n=0,8,for(k=0,(n+1)*(n+2)/2-1,print1(T(n,k),","));print("")) */}

Formula

T(n,n) = A000108(n);

A009766 (Catalan's triangle) forms lower left sub-triangle;

T(n+1,2*n+1) = A127632(n), where g.f. of A127632 is: 2/(1+sqrt(2*sqrt(1-4*x)-1)).

T(n,n*(n+1)/2) = A127716(n).

T(n,(n+1)*(n+2)/2-1) = A127715(n).