A156763 Triangle T(n, k) = binomial(2*k, k)*binomial(n+k, n-k) + binomial(2*n-k, k)*binomial(2*(n-k), n-k), read by rows.

2, 3, 3, 7, 12, 7, 21, 42, 42, 21, 71, 160, 180, 160, 71, 253, 660, 770, 770, 660, 253, 925, 2814, 3570, 3360, 3570, 2814, 925, 3433, 12068, 17388, 15750, 15750, 17388, 12068, 3433, 12871, 51552, 85344, 81312, 69300, 81312, 85344, 51552, 12871

Offset: 0

Author

Roger L. Bagula, Feb 15 2009

Keywords

• nonn
• tabl

Examples

			Triangle begins as:
      2;
      3,      3;
      7,     12,      7;
     21,     42,     42,     21;
     71,    160,    180,    160,     71;
    253,    660,    770,    770,    660,    253;
    925,   2814,   3570,   3360,   3570,   2814,    925;
   3433,  12068,  17388,  15750,  15750,  17388,  12068,   3433;
  12871,  51552,  85344,  81312,  69300,  81312,  85344,  51552,  12871;
  48621, 218880, 413820, 438900, 342342, 342342, 438900, 413820, 218880, 48621;
		

References

• J. Riordan, Combinatorial Identities, Wiley, 1968, p. 66.

Links

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

Crossrefs

Cf. A001850, A063007.

Programs

Magma

A063007:= func< n,k | Binomial(n, k)*Binomial(n+k, k) >;
A156763:= func< n,k | A063007(n,k) + A063007(n,n-k) >;
[A156763(n,k): k in [0..n]. n in [0..12]]; // G. C. Greubel, Jun 15 2021

Mathematica

T[n_, k_]:= Binomial[n+k, n-k]*Binomial[2*k, k] + Binomial[2*(n-k), n-k]*Binomial[ 2*n-k, k];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Jun 15 2021 *)

Sage

def A063007(n, k): return binomial(n+k, n-k)*binomial(2*k, k)
def A156763(n, k): return A063007(n,k) + A063007(n,n-k)
flatten([[A156763(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 15 2021

Formula

T(n, k) = binomial(2*k, k)*binomial(n+k, n-k) + binomial(2*n-k, k)*binomial(2*(n-k), n-k).

T(n, k) = A063007(n, k) + A063007(n, n-k).

Sum_{k=0..n} T(n, k) = 2*A001850(n). - G. C. Greubel, Jun 15 2021

Extensions

Edited by G. C. Greubel, Jun 15 2021