A127774 Triangle read by rows: row n consists of n-1 zeros followed by A000292(n).

1, 0, 4, 0, 0, 10, 0, 0, 0, 20, 0, 0, 0, 0, 35, 0, 0, 0, 0, 0, 56, 0, 0, 0, 0, 0, 0, 84, 0, 0, 0, 0, 0, 0, 0, 120, 0, 0, 0, 0, 0, 0, 0, 0, 165, 0, 0, 0, 0, 0, 0, 0, 0, 0, 220, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 286, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 364

Offset: 1

Author

Gary W. Adamson, Jan 28 2007

Keywords

• nonn
• tabl
• easy

Comments

Essentially triangle T(n,k), read by rows, given by (0,0,0,0,0,0,0,...) DELTA (4,-3/2,5/6,-1/3,3/5,-1/10,1/2,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 14 2011

Examples

			First few rows of the triangle are:
  1;
  0,  4;
  0,  0, 10;
  0,  0,  0, 20;
  0,  0,  0,  0, 35;
  ...
		

Crossrefs

Cf. A000292, A127648, A127773.

Programs

Python

from math import isqrt
from sympy.ntheory.primetest import is_square
def A127774(n): return (a:=(m:=isqrt(k:=n<<1))+(k>m*(m+1)))*(a+1)*(a+2)//6 if is_square((n<<3)+1) else 0 # Chai Wah Wu, Jun 09 2025

Formula

G.f.: 1/((x*y-1)^4). - R. J. Mathar, Aug 12 2015

Extensions

More terms from Michel Marcus, Jun 10 2025