A123736 Triangle T(n,k) = Sum_{j=0..k/2} binomial(n-j-1,k-2*j), read by rows.

1, 0, 1, 1, 1, 0, 1, 2, 2, 1, 1, 0, 1, 3, 4, 3, 2, 1, 1, 0, 1, 4, 7, 7, 5, 3, 2, 1, 1, 0, 1, 5, 11, 14, 12, 8, 5, 3, 2, 1, 1, 0, 1, 6, 16, 25, 26, 20, 13, 8, 5, 3, 2, 1, 1, 0, 1, 7, 22, 41, 51, 46, 33, 21, 13, 8, 5, 3, 2, 1, 1, 0, 1, 8, 29, 63, 92, 97, 79, 54, 34, 21, 13, 8, 5, 3, 2, 1, 1, 0, 1

Offset

1,8

Comments

Row sums give: A000225

Links

G. C. Greubel, Rows n = 1..100 of triangle, flattened

Eric Weisstein's World of Mathematics, Steenrod Algebra

Example

The triangle starts in row n=1 with columns 0 <= k < 2*n:
  1, 0;
  1, 1,  1,  0;
  1, 2,  2,  1,  1,  0;
  1, 3,  4,  3,  2,  1,  1,  0;
  1, 4,  7,  7,  5,  3,  2,  1,  1,  0;
  1, 5, 11, 14, 12,  8,  5,  3,  2,  1,  1, 0;
  1, 6, 16, 25, 26, 20, 13,  8,  5,  3,  2, 1, 1, 0;
  1, 7, 22, 41, 51, 46, 33, 21, 13,  8,  5, 3, 2, 1, 1, 0;
  1, 8, 29, 63, 92, 97, 79, 54, 34, 21, 13, 8, 5, 3, 2, 1, 1, 0;

Maple

seq(seq(sum(binomial(n-j-1, k-2*j), j=0..floor(k/2)), k=0..2*n-1), n=1..10); # G. C. Greubel, Sep 05 2019

Mathematica

Table[Sum[Binomial[n-j-1, k-2*j], {j, 0, Floor[k/2]}], {n, 10}, {k, 0, 2*n-1}]//Flatten (* modified by G. C. Greubel, Sep 05 2019 *)

Prog

(PARI) T(n, k) = sum(j=0, k\2, binomial(n-j-1, k-2*j));
for(n=1, 10, for(k=0, 2*n-1, print1(T(n, k), ", "))) \\ G. C. Greubel, Sep 05 2019
(Magma) [&+[Binomial(n-j-1, k-2*j): j in [0..Floor(k/2)]]: k in [0..2*n-1], n in [1..10]]; // G. C. Greubel, Sep 05 2019
(Sage) [[sum(binomial(n-j-1, k-2*j) for j in (0..floor(k/2))) for k in (0..2*n-1)] for n in (1..10)] # G. C. Greubel, Sep 05 2019
(GAP) Flat(List([1..10], n-> List([0..2*n-1], k-> Sum([0..Int(k/2)], j-> Binomial(n-j-1, k-2*j) )))); # G. C. Greubel, Sep 05 2019

Crossrefs

Cf. A136431 (antidiagonals), A027926 (row-reversed), A004006 (column m=3)

Sequence in context: A265890 A226920 A362759 * A358476 A185304 A081389

Adjacent sequences: A123733 A123734 A123735 * A123737 A123738 A123739

Keyword

nonn,easy,tabf

Author

Roger L. Bagula, Nov 14 2006

Status

approved