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

0,1

Links

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

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

Example

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;

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

Prog

(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
(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

Crossrefs

Row sums are A000984(n+1).

Cf. A001700, A007318, A054142, A085478.

Sequence in context: A185909 A347533 A193713 * A143444 A108346 A210558

Adjacent sequences: A171821 A171822 A171823 * A171825 A171826 A171827

Keyword

nonn,tabl,easy

Author

Roger L. Bagula, Dec 19 2009

Extensions

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

Status

approved