A159841 Triangle T(n,k) = binomial(3*n+1, 2*n+k+1), read by rows.

1, 4, 1, 21, 7, 1, 120, 45, 10, 1, 715, 286, 78, 13, 1, 4368, 1820, 560, 120, 16, 1, 27132, 11628, 3876, 969, 171, 19, 1, 170544, 74613, 26334, 7315, 1540, 231, 22, 1, 1081575, 480700, 177100, 53130, 12650, 2300, 300, 25, 1, 6906900, 3108105, 1184040, 376740

Offset

0,2

Comments

T(n,0) = A045721(n), T(2n,n) = A079590(n).

Links

G. C. Greubel, Rows n=0..100 of triangle, flattened

E. H. M. Brietzke, An identity of Andrews and a new method for the Riordan array proof of combinatorial identities, Discrete Math., 308 (2008), 4246-4262.

Formula

T(n,0) = 4*T(n-1,0) + 5*T(n-1,1) + T(n-1,2), T(n+1,k+1) = T(n,k) + 3*T(n,k+1) + 3*T(n,k+2) + T(n,k+3) for k >= 0.

Example

Triangle begins:
     1;
     4,    1;
    21,    7,    1;
   120,   45,   10,    1;
   715,  286,   78,   13,    1;
  4368, 1820,  560,  120,   16,    1;
  ...

Mathematica

f[n_, k_]:=Binomial[3n+1, 2n+k+1]; Table[ f[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Robert G. Wilson v, May 31 2009 *)

Prog

(PARI) for(n=0, 10, for(k=0, n, print1(binomial(3*n+1, 2*n+k+1), ", "))) \\ G. C. Greubel, May 19 2018
(Magma) /* As triangle */ [[Binomial(3*n+1, 2*n+k+1): k in [0..n]]: n in [0..10]]; // G. C. Greubel, May 19 2018

Crossrefs

Cf. A045721, A079590.

Sequence in context: A144484 A121336 A126457 * A202550 A364760 A142472

Adjacent sequences: A159838 A159839 A159840 * A159842 A159843 A159844

Keyword

nonn,tabl

Author

Philippe Deléham, Apr 23 2009

Extensions

More terms from Robert G. Wilson v, May 31 2009

Status

approved