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

0,1

References

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

Links

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

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

Example

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;

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 *)

Prog

(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
(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

Crossrefs

Cf. A001850, A063007.

Sequence in context: A167886 A095978 A370361 * A169653 A129012 A136122

Adjacent sequences: A156760 A156761 A156762 * A156764 A156765 A156766

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 15 2009

Extensions

Edited by G. C. Greubel, Jun 15 2021

Status

approved