A081181 Staircase on Pascal's triangle.

1, 3, 6, 20, 35, 126, 210, 792, 1287, 5005, 8008, 31824, 50388, 203490, 319770, 1307504, 2042975, 8436285, 13123110, 54627300, 84672315, 354817320, 548354040, 2310789600, 3562467300, 15084504396, 23206929840, 98672427616, 151532656696

Offset: 0

Author

Paul Barry, Mar 11 2003

Keywords

• easy
• nonn

Comments

Arrange Pascal's triangle as a square array. a(n) is then a diagonal staircase on the square array. A companion staircase is given by A065942.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A065942.

Programs

Magma

[Binomial(Floor((n+1)/2)+(n+1), n): n in [0..30]]; // Vincenzo Librandi, Aug 06 2013

Mathematica

Table[Binomial[Floor[(n + 1) / 2] + (n + 1), n], {n, 0, 30}] (* Vincenzo Librandi, Aug 06 2013 *)

SageMath

[binomial(((n+1)//2)+(n+1), n) for n in range(41)] # G. C. Greubel, Jan 14 2024

Formula

a(n) = binomial(floor((n + 1)/2) + (n + 1), n).

Conjecture: 8*n*(n+3)*(1845*n-2882)*a(n) +4*(-5097*n^3+11143*n^2 +42110*n-27416)*a(n-1) +6*(-16605*n^3-7272*n^2-16701*n+9490)*a(n-2) +3*(3*n-5)*(5097*n-949)*(3*n-4)*a(n-3)=0. - R. J. Mathar, Oct 29 2014