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

Author

Philippe Deléham, Apr 23 2009

Keywords

• nonn
• tabl

Comments

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

Examples

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

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.

Crossrefs

Cf. A045721, A079590.

Programs

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

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 *)

PARI

for(n=0,10, for(k=0,n, print1(binomial(3*n+1, 2*n+k+1), ", "))) \\ G. C. Greubel, May 19 2018

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.

Extensions

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