A123225 Triangle read by rows: T(n,d) = (n!/d!)*(n+1)*binomial(2n-d+1,n+1)/(n-d+1) (0 <= d <= n).

1, 3, 2, 20, 12, 3, 210, 120, 30, 4, 3024, 1680, 420, 60, 5, 55440, 30240, 7560, 1120, 105, 6, 1235520, 665280, 166320, 25200, 2520, 168, 7, 32432400, 17297280, 4324320, 665280, 69300, 5040, 252, 8, 980179200, 518918400, 129729600, 20180160, 2162160

Offset

0,2

Links

G. C. Greubel, Rows n = 0..100 of triangle, flattened

E. Babson and E. Steingrimsson, Generalized permutation patterns and a classification of the Mahonian statistics, Séminaire Lotharingien de Combinatoire, Paper B44b, 2000 (see p. 8).

Formula

T(n,d) = (n!/d!)*Sum_{k=0..n} binomial(n-d+k,k) (0 <= d <= n).

T(n,d) = (n!/d!)*(n+1)*binomial(2n-d+1,n+1)/(n-d+1) (0 <= d <= n).

Example

Triangle begins:
     1;
     3,    2;
    20,   12,   3;
   210,  120,  30,  4;
  3024, 1680, 420, 60, 5;

Maple

T:=proc(n, d) if d<=n then n!*(n+1)*binomial(2*n-d+1, n+1)/d!/(n-d+1) else 0 fi end: for n from 0 to 9 do seq(T(n, d), d=0..n) od; # yields sequence in triangular form

Mathematica

T[n_, d_, k_] = (n!/d!)*Binomial[n - d + k, k]; a = Table[Table[Sum[T[n, d, k], {k, 0, n}], {d, 0, n}], {n, 0, 10}]; Flatten[a]

Prog

(PARI) for(n=0, 12, for(k=0, n, print1((n!/k!)*(n+1)*binomial(2*n-k+1, n+1)/(n-k+1), ", "))) \\ G. C. Greubel, Oct 12 2018
(Magma) [[(Factorial(n)/Factorial(k))*(n+1)*Binomial(2*n-k+1, n+1)/(n - k +1): k in [0..n]]: n in [0..12]]; // G. C. Greubel, Oct 12 2018
(GAP) Flat(List([0..8], n->List([0..n], d->(Factorial(n)/Factorial(d))*(n+1)*Binomial(2*n-d+1, n+1)/(n-d+1)))); # Muniru A Asiru, Oct 12 2018

Crossrefs

Cf. A008302.

Sequence in context: A154261 A098655 A065038 * A009028 A009022 A009033

Adjacent sequences: A123222 A123223 A123224 * A123226 A123227 A123228

Keyword

nonn,tabl

Author

Roger L. Bagula, Oct 05 2006

Extensions

Edited by N. J. A. Sloane, Oct 08 2006

Status

approved