A156052 Triangle read by rows: T(n, k) = binomial(n, k)/Beta(n+1, n-k+1) + binomial(n, n-k)/Beta(n+1, k+1).

2, 8, 8, 33, 48, 33, 144, 240, 240, 144, 635, 1240, 1260, 1240, 635, 2778, 6510, 6720, 6720, 6510, 2778, 12019, 33600, 38430, 33600, 38430, 33600, 12019, 51488, 168672, 223776, 184800, 184800, 223776, 168672, 51488, 218799, 824400, 1275120, 1119888, 900900, 1119888, 1275120, 824400, 218799

Offset: 0

Author

Roger L. Bagula, Feb 02 2009

Keywords

• nonn
• tabl

Comments

Row sums are 2*A108666(n+1): {2, 16, 114, 768, 5010, 32016, 201698, 1257472, 7777314, 47800080, 292292946, ...}.

Examples

			Triangle begins as:
      2;
      8,      8;
     33,     48,     33;
    144,    240,    240,    144;
    635,   1240,   1260,   1240,    635;
   2778,   6510,   6720,   6720,   6510,   2778;
  12019,  33600,  38430,  33600,  38430,  33600,  12019;
  51488, 168672, 223776, 184800, 184800, 223776, 168672, 51488;
		

Links

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

Programs

GAP

B:=Binomial;; Flat(List([0..10], n-> List([0..n], k-> (k+1)*B(n+1, k+1)*( B(2*n-k+1, n+1) + B(n+k+1, n+1) ) ))); # G. C. Greubel, Dec 01 2019

Magma

B:=Binomial; [(k+1)*B(n+1, k+1)*( B(2*n-k+1, n+1) + B(n+k+1, n+1) ): k in [0..n], n in [0..10]]; // G. C. Greubel, Dec 01 2019

Maple

b:=binomial; seq(seq( (k+1)*b(n+1, k+1)*( b(2*n-k+1, n+1) + b(n+k+1, n+1) ), k=0..n), n=0..10); # G. C. Greubel, Dec 01 2019

Mathematica

Table[Binomial[n, k]/Beta[n+1, n-k+1] + Binomial[n, n-k]/Beta[n+1, k+1], {n, 0, 10}, {k, 0, n}]//FlattenTable[(k+1)*Binomial[n+1, k+1]*(Binomial[n+k+1, n+1] + Binomial[2*n-k+1, n+1]), {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Dec 01 2019 *)

PARI

T(n, k) = my(b=binomial); (k+1)*b(n+1, k+1)*( b(2*n-k+1, n+1) + b(n+k+1, n+1) ); \\ G. C. Greubel, Dec 01 2019

Sage

b=binomial; [[(k+1)*b(n+1, k+1)*( b(2*n-k+1, n+1) + b(n+k+1, n+1) ) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Dec 01 2019

Formula

T(n, k) = binomial(n, k)/Beta(n+1, n-k+1) + binomial(n, n-k)/Beta(n+1, k+1).

T(n, k) = (k+1)*binomial(n+1, k+1)*( binomial(2*n-k+1, n+1) + binomial(n+k+1, n+1) ). - G. C. Greubel, Dec 01 2019