A121757 Triangle read by rows: multiply Pascal's triangle by 1,2,6,24,120,720,... = A000142.

1, 1, 2, 1, 4, 6, 1, 6, 18, 24, 1, 8, 36, 96, 120, 1, 10, 60, 240, 600, 720, 1, 12, 90, 480, 1800, 4320, 5040, 1, 14, 126, 840, 4200, 15120, 35280, 40320, 1, 16, 168, 1344, 8400, 40320, 141120, 322560, 362880, 1, 18, 216, 2016, 15120, 90720, 423360, 1451520

Offset

0,3

Comments

Row sums are 1,3,11,49,261,1631,... = A001339

a(n,k) = D(n+1,k+1) Array D in A253938 is part of a conjectured formula for F(n,p,r) that relates Dyck path peaks and returns. a(n,k) was discovered prior to array D. - Roger Ford, May 19 2016

Links

Vincenzo Librandi, Rows n = 0..100, flattened

J. Goldman, J. Haglund, Generalized rook polynomials, J. Combin. Theory A91 (2000), 509-530, 2-rook coefficients of k rooks on the 2xn board (all heights 2).

Index entries for triangles and arrays related to Pascal's triangle

Formula

a(n,k) = A007318(n,k)*A000142(k+1), k=0,1,..,n, n=0,1,2,3... - R. J. Mathar, Sep 02 2006

a(n,k) = A008279(n,k) * (k+1). a(n,k) = n!*(k+1)/(n-k)!. - Franklin T. Adams-Watters, Sep 20 2006

Example

Row 6 is 1*1 5*2 10*6 10*24 5*120 1*720.
From Vincenzo Librandi, Dec 16 2012: (Start)
Triangle begins:
1,
1, 2,
1, 4,  6,
1, 6,  18,  24,
1, 8,  36,  96,   120,
1, 10, 60,  240,  600,  720,
1, 12, 90,  480,  1800, 4320,  5040,
1, 14, 126, 840,  4200, 15120, 35280,  40320,
1, 16, 168, 1344, 8400, 40320, 141120, 322560, 362880 etc.
(End)

Mathematica

Flatten[Table[n!(k+1)/(n-k)!, {n, 0, 10}, {k, 0, n}]]  (* Harvey P. Dale, Apr 25 2011 *)

Prog

(PARI) A000142(n)={ return(n!) ; } A007318(n, k)={ return(binomial(n, k)) ; } A121757(n, k)={ return(A007318(n, k)*A000142(k+1)) ; } { for(n=0, 12, for(k=0, n, print1(A121757(n, k), ", ") ; ); ) ; } \\ R. J. Mathar, Sep 02 2006
(Haskell)
a121757 n k = a121757_tabl !! n !! k
a121757_row n = a121757_tabl !! n
a121757_tabl = iterate
   (\xs -> zipWith (+) (xs ++ [0]) (zipWith (*) [1..] ([0] ++ xs))) [1]
-- Reinhard Zumkeller, Mar 06 2014

Crossrefs

Cf. A007526 A000522, A005843 (2nd column), A028896 (3rd column).

Cf. A008279.

Cf. A008277, A132159 (mirrored).

Sequence in context: A208765 A232335 A098473 * A219441 A219142 A220226

Adjacent sequences: A121754 A121755 A121756 * A121758 A121759 A121760

Keyword

nonn,easy,tabl

Author

Alford Arnold, Aug 19 2006

Status

approved