A156367 Triangle T(n, k) = binomial(n+k, 2*k)*k!, read by rows.

1, 1, 1, 1, 3, 2, 1, 6, 10, 6, 1, 10, 30, 42, 24, 1, 15, 70, 168, 216, 120, 1, 21, 140, 504, 1080, 1320, 720, 1, 28, 252, 1260, 3960, 7920, 9360, 5040, 1, 36, 420, 2772, 11880, 34320, 65520, 75600, 40320, 1, 45, 660, 5544, 30888, 120120, 327600, 604800, 685440, 362880

Offset: 0

Author

Paul Barry, Feb 08 2009

Keywords

• easy
• nonn
• tabl

Examples

			Triangle begins
  1;
  1,  1;
  1,  3,   2;
  1,  6,  10,    6;
  1, 10,  30,   42,    24;
  1, 15,  70,  168,   216,   120;
  1, 21, 140,  504,  1080,  1320,   720;
  1, 28, 252, 1260,  3960,  7920,  9360,  5040;
  1, 36, 420, 2772, 11880, 34320, 65520, 75600, 40320;
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A084261 (diagonal sums), A155856 (row reversal), A155857 (row sums)

Programs

Mathematica

Flatten[Table[Binomial[n+k,2k]k!,{n,0,10},{k,0,n}]] (* Harvey P. Dale, Jun 17 2015 *)

Sage

flatten([[factorial(k)*binomial(n+k, 2*k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 05 2021

Formula

G.f.: 1/(1 -x -x*y/(1 -x -x*y/(1 -x -2*x*y/(1 -x -2*x*y/(1 -x -3*x*y/(1 -x -3*x*y/(1 - ... (continued fraction).

T(n, k) = binomial(n+k, 2*k)*k!

T(n, k) = A155856(n, n-k).

Sum_{k=0..n} T(n, k) = A155857(n).

sum_{k=0..floor(n/2)} T(n, k) = A084261(n).