A071944 Triangle read by rows giving numbers of paths in a lattice satisfying certain conditions.

1, 1, 1, 1, 2, 2, 1, 3, 5, 6, 1, 4, 9, 16, 19, 1, 5, 14, 31, 54, 63, 1, 6, 20, 52, 111, 188, 219, 1, 7, 27, 80, 197, 405, 676, 787, 1, 8, 35, 116, 320, 752, 1508, 2492, 2897, 1, 9, 44, 161, 489, 1276, 2900, 5712, 9361, 10869, 1, 10, 54, 216, 714, 2034, 5095, 11296, 21933, 35702, 41414

Offset: 0

Author

N. J. A. Sloane, Jun 15 2002

Keywords

• nonn
• easy
• tabl

Examples

			Triangle begins with:
  1;
  1,   1;
  1,   2,   2;
  1,   3,   5,   6;
  1,   4,   9,  16,  19;
  1,   5,  14,  31,  54,  63;
  1,   6,  20,  52, 111, 188, 219;
  1,   7,  27,  80, 197, 405, 676, 787;
  ...
		

Links

• G. C. Greubel, Rows n = 0..100 of triangle, flattened
• D. Merlini, D. G. Rogers, R. Sprugnoli and M. C. Verri, On some alternative characterizations of Riordan arrays, Canad J. Math., 49 (1997), 301-320.

Crossrefs

Diagonal entries form A071969.

Programs

Magma

[[((n-k+1)/(n+1))*(&+[Binomial(n+1, j)*Binomial(n+k -3*j, n): j in [0..Floor(k/3)]]): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Mar 17 2019

Maple

a := proc(n,k) if k<=n then (n-k+1)*sum(binomial(n+1,i)*binomial(n+k-3*i,n),i=0..k/3)/(n+1) else 0 fi end;

Mathematica

Table[((n-k+1)/(n+1))*Sum[Binomial[n+1, j]*Binomial[n+k-3*j, n], {j, 0, k/3}], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 17 2019 *)

PARI

{T(n,k) = ((n-k+1)/(n+1))*sum(j=0, floor(k/3), binomial(n+1, j)* binomial(n+k -3*j, n))}; \\ G. C. Greubel, Mar 17 2019

Sage

[[((n-k+1)/(n+1))*sum(binomial(n+1,j)*binomial(n+k-3*j,n) for j in (0..floor(k/3))) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Mar 17 2019

Formula

T(n, k) = ((n-k+1)/(n+1))*Sum_{i=0..k/3} binomial(n+1, i)*binomial(n+k -3*i, n), for k <= n.

Extensions

More terms from Emeric Deutsch, Dec 19 2003