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

0,1

Links

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

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)

Example

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;

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

Prog

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

Crossrefs

Cf. A000984, A036910, A323230.

Sequence in context: A308513 A045683 A343031 * A155755 A080088 A098715

Adjacent sequences: A157528 A157529 A157530 * A157532 A157533 A157534

Keyword

nonn,tabl

Author

Roger L. Bagula, Mar 02 2009

Extensions

Edited by G. C. Greubel, Dec 09 2021

Status

approved