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

0,5

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.

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.

Example

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;
  ...

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 *)

Prog

(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
(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
(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

Crossrefs

Diagonal entries form A071969.

Sequence in context: A059718 A076038 A095788 * A309495 A080955 A340108

Adjacent sequences: A071941 A071942 A071943 * A071945 A071946 A071947

Keyword

nonn,easy,tabl

Author

N. J. A. Sloane, Jun 15 2002

Extensions

More terms from Emeric Deutsch, Dec 19 2003

Status

approved