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

0,2

Links

Table of n, a(n) for n=0..44.

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.

Example

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,
  ...

Prog

(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

Crossrefs

Lower triangle of the square array A324010.

Sequence in context: A276268 A350081 A350080 * A086909 A046963 A022386

Adjacent sequences: A306684 A306685 A306686 * A306688 A306689 A306690

Keyword

nonn,easy,tabl

Author

Günter Rote, Mar 05 2019

Status

approved