A157531 Triangle T(n, k) = binomial(2*n, n) + binomial(n, k)^2, read by rows.

2, 3, 3, 7, 10, 7, 21, 29, 29, 21, 71, 86, 106, 86, 71, 253, 277, 352, 352, 277, 253, 925, 960, 1149, 1324, 1149, 960, 925, 3433, 3481, 3873, 4657, 4657, 3873, 3481, 3433, 12871, 12934, 13654, 16006, 17770, 16006, 13654, 12934, 12871, 48621, 48701, 49916, 55676, 64496, 64496, 55676, 49916, 48701, 48621

Offset: 0

Author

Roger L. Bagula, Mar 02 2009

Keywords

• nonn
• tabl

Examples

			Triangle begins as:
      2;
      3,     3;
      7,    10,     7;
     21,    29,    29,    21;
     71,    86,   106,    86,    71;
    253,   277,   352,   352,   277,   253;
    925,   960,  1149,  1324,  1149,   960,   925;
   3433,  3481,  3873,  4657,  4657,  3873,  3481,  3433;
  12871, 12934, 13654, 16006, 17770, 16006, 13654, 12934, 12871;
  48621, 48701, 49916, 55676, 64496, 64496, 55676, 49916, 48701, 48621;
		

Links

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

Crossrefs

Cf. A000984, A036910, A323230.

Programs

Magma

[Binomial(2*n, n) + Binomial(n, k)^2: k in [0..n], n in [0..12]]; // G. C. Greubel, Dec 09 2021

Maple

A157531 := proc(n,k)
    binomial(2*n,n)+binomial(n,k)^2 ;
end proc:
seq(seq(A157531(n,k),k=0..n),n=0..12) ; # R. J. Mathar, Jan 12 2023

Mathematica

T[n_, k_]:= T[n,k]= Sum[Binomial[n, j]^2, {j,0,k}] + Sum[Binomial[n, j]^2, {j, 0, n-k}];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten

Sage

flatten([[binomial(2*n, n) + binomial(n, k)^2 for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Dec 09 2021

Formula

T(n, k) = Sum_{j=0..k} binomial(n,j)*binomial(n,n-j) + Sum_{j=0..n-k} binomial(n,j)*binomial(n,n-j).

From G. C. Greubel, Dec 09 2021: (Start)

Sum_{k=0..n} T(n, k) = (n+2)*binomial(2*n, n).

T(n, k) = T(n, n-k).

T(n, 0) = 1 + binomial(2*n, n) = A323230(n+1).

T(2*n, n) = 2*A036910(n). (End)

Extensions

Edited by G. C. Greubel, Dec 09 2021