A118394 Triangle T(n,k) = n!/(k!*(n-3*k)!), for n >= 3*k >= 0, read by rows.

1, 1, 1, 1, 6, 1, 24, 1, 60, 1, 120, 360, 1, 210, 2520, 1, 336, 10080, 1, 504, 30240, 60480, 1, 720, 75600, 604800, 1, 990, 166320, 3326400, 1, 1320, 332640, 13305600, 19958400, 1, 1716, 617760, 43243200, 259459200, 1, 2184, 1081080, 121080960, 1816214400

Offset

0,5

Comments

Row sums form A118395.

Eigenvector is A118396.

Links

G. C. Greubel, Rows n = 0..150 of the triangle, flattened

Formula

E.g.f.: A(x,y) = exp(x + y*x^3).

Example

Triangle begins:
  1;
  1;
  1;
  1,    6;
  1,   24;
  1,   60;
  1,  120,    360;
  1,  210,   2520;
  1,  336,  10080;
  1,  504,  30240,    60480;
  1,  720,  75600,   604800;
  1,  990, 166320,  3326400;
  1, 1320, 332640, 13305600, 19958400;
  ...

Mathematica

T[n_, k_] := n!/(k!(n-3k)!);
Table[T[n, k], {n, 0, 14}, {k, 0, Floor[n/3]}] // Flatten (* Jean-François Alcover, Nov 04 2020 *)

Prog

(PARI) T(n, k)=if(n<3*k || k<0, 0, n!/k!/(n-3*k)!)
(Sage)
f=factorial;
flatten([[f(n)/(f(k)*f(n-3*k)) for k in [0..n/3]] for n in [0..20]]) # G. C. Greubel, Mar 07 2021
(Magma)
F:= Factorial;
[F(n)/(F(k)*F(n-3*k)): k in [0..Floor(n/3)], n in [0..20]]; // G. C. Greubel, Mar 07 2021

Crossrefs

Cf. A118395 (row sums), A118396 (eigenvector).

Variants: A059344, A118931.

Sequence in context: A050300 A185678 A286893 * A365372 A278906 A281517

Adjacent sequences: A118391 A118392 A118393 * A118395 A118396 A118397

Keyword

nonn,tabf

Author

Paul D. Hanna, May 07 2006

Status

approved