A306687 Triangular array read by rows: The sum of squares of the number of common points in all pairs of lattice paths from (0,0) to (x,y), for 0 <= y <= x (the unnormalized second moment).

1, 4, 26, 9, 92, 474, 16, 240, 1704, 8084, 25, 520, 4879, 29560, 134450, 36, 994, 11928, 89928, 498140, 2208612, 49, 1736, 25956, 238440, 1580810, 8265432, 36024884, 64, 2832, 51648, 568128, 4442768, 27055808, 135873360, 584988840, 81, 4380, 95733, 1242648, 11320595, 79443000, 455434875, 2220096240, 9470766690

Offset: 0

Author

Günter Rote, Mar 05 2019

Keywords

• nonn
• easy
• tabl

Examples

			T(1,1) = 26, because the two lattice paths are DR and RD. (DR,DR) and (RD,RD) have three common points, (DR,RD) and (RD,DR) have two common points, and 2*3^2+2*2^2 = 26. - _Charlie Neder_, Jun 26 2019
The triangle begins:
   1,
   4,  26,
   9,  92,  474,
  16, 240, 1704,  8084,
  25, 520, 4879, 29560, 134450,
  ...
		

Crossrefs

Lower triangle of the square array A324010.

Programs

PARI

a(x,y) = (x+y+1)*binomial(x+y+2,x+1)*binomial(x+y,x)-binomial(2*x+2*y+2,2*x+1)/2;
for (n=0, 10, for (k=0, n, print1(a(n,k), ", ")); print) \\ Michel Marcus, Apr 08 2019

Formula

T(x,y) = (x+y+1) * binomial(x+y+2,x+1) * binomial(x+y,x) - binomial(2*x+2*y+2,2*x+1) / 2.