A171824 Triangle T(n,k)= binomial(n + k,n) + binomial(2*n-k,n) read by rows.

2, 3, 3, 7, 6, 7, 21, 14, 14, 21, 71, 40, 30, 40, 71, 253, 132, 77, 77, 132, 253, 925, 469, 238, 168, 238, 469, 925, 3433, 1724, 828, 450, 450, 828, 1724, 3433, 12871, 6444, 3048, 1452, 990, 1452, 3048, 6444, 12871, 48621, 24320, 11495, 5225, 2717, 2717, 5225, 11495, 24320, 48621

Offset: 0

Author

Roger L. Bagula, Dec 19 2009

Keywords

• nonn
• tabl
• easy

Examples

			Triangle begins as:
       2;
       3,     3;
       7,     6,     7;
      21,    14,    14,    21;
      71,    40,    30,    40,   71;
     253,   132,    77,    77,  132,  253;
     925,   469,   238,   168,  238,  469, 925;
    3433,  1724,   828,   450,  450,  828, 1724,  3433;
   12871,  6444,  3048,  1452,  990, 1452, 3048,  6444, 12871;
   48621, 24320, 11495,  5225, 2717, 2717, 5225, 11495, 24320, 48621;
  184757, 92389, 43824, 19734, 9009, 6006, 9009, 19734, 43824, 92389, 184757;
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1325

Crossrefs

Row sums are A000984(n+1).

Cf. A001700, A007318, A054142, A085478.

Programs

Magma

T:= func< n,k | Binomial(n+k,n) + Binomial(2*n-k,n) >;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 29 2021

Mathematica

T[n_, k_] = Binomial[n+k, k] + Binomial[2*n-k, n-k];
Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten

Sage

def T(n, k): return binomial(n+k,n) + binomial(2*n-k,n)
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 29 2021

Formula

T(n,k) = A046899(n,k) + A092392(n,k).

Sum_{k=0..n} T(n,k) = binomial(2*n+2, n+1) = 2*A001700(n) = A000984(n+1). - G. C. Greubel, Apr 29 2021

Extensions

Formula and row sums reference added by the Assoc. Editors of the OEIS, Feb 24 2010