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

Author

Roger L. Bagula, Nov 14 2006

Keywords

• nonn
• easy
• tabf

Comments

Row sums give: A000225

Examples

			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;
		

Links

• G. C. Greubel, Rows n = 1..100 of triangle, flattened
• Eric Weisstein's World of Mathematics, Steenrod Algebra

Crossrefs

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

Programs

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

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

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 *)

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

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