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

0,5

Links

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

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).

Example

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;

Mathematica

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

Prog

(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

Crossrefs

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

Sequence in context: A088617 A190909 A144250 * A193593 A308616 A181853

Adjacent sequences: A156364 A156365 A156366 * A156368 A156369 A156370

Keyword

easy,nonn,tabl

Author

Paul Barry, Feb 08 2009

Status

approved