A173881 Triangle T(n,k) = k*binomial(n,k)*binomial(n-1,k) with T(n,0) = T(n,n) = 1, read by rows.

1, 1, 1, 1, 2, 1, 1, 6, 6, 1, 1, 12, 36, 12, 1, 1, 20, 120, 120, 20, 1, 1, 30, 300, 600, 300, 30, 1, 1, 42, 630, 2100, 2100, 630, 42, 1, 1, 56, 1176, 5880, 9800, 5880, 1176, 56, 1, 1, 72, 2016, 14112, 35280, 35280, 14112, 2016, 72, 1, 1, 90, 3240, 30240, 105840, 158760, 105840, 30240, 3240, 90, 1

Offset

0,5

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Formula

T(n, k) = c(n)/(c(k)*c(n-k)) where c(n) = Product_{j=2..n} j*(j-1) with c(0) = c(1) = 1.

From G. C. Greubel, Apr 17 2021: (Start)

T(n,k) = k*binomial(n,k)*binomial(n-1,k) with T(n,0) = T(n,n) = 1.

Sum_{k=0..n} T(n,k) = 2*(2*n-3)*binomial(2*n-4, n-2) + 2 - [n=0] = 2 + 2A002457(n-2) - [n=0]. (End)

Example

Triangle begins as:
  1;
  1,  1;
  1,  2,    1;
  1,  6,    6,     1;
  1, 12,   36,    12,      1;
  1, 20,  120,   120,     20,      1;
  1, 30,  300,   600,    300,     30,      1;
  1, 42,  630,  2100,   2100,    630,     42,     1;
  1, 56, 1176,  5880,   9800,   5880,   1176,    56,    1;
  1, 72, 2016, 14112,  35280,  35280,  14112,  2016,   72,  1;
  1, 90, 3240, 30240, 105840, 158760, 105840, 30240, 3240, 90, 1;
  ...

Mathematica

T[n_, k_]:= If[k==0 || k==n, 1, k*Binomial[n, k]*Binomial[n-1, k]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Apr 17 2021 *)

Prog

(Magma)
T:= func< n, k | k eq 0 or k eq n select 1 else k*Binomial(n, k)*Binomial(n-1, k) >;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 17 2021
(Sage)
def T(n, k): return 1 if (k==0 or k==n) else k*binomial(n, k)*binomial(n-1, k)
flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 17 2021

Crossrefs

Sequence in context: A155864 A145903 A223257 * A329228 A172373 A174411

Adjacent sequences: A173878 A173879 A173880 * A173882 A173883 A173884

Keyword

nonn,tabl

Author

Roger L. Bagula, Mar 01 2010

Extensions

Edited by G. C. Greubel, Apr 17 2021

Status

approved