A099573 Reverse of number triangle A054450.

1, 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 4, 5, 1, 1, 5, 5, 8, 8, 1, 1, 6, 6, 12, 12, 13, 1, 1, 7, 7, 17, 17, 21, 21, 1, 1, 8, 8, 23, 23, 33, 33, 34, 1, 1, 9, 9, 30, 30, 50, 50, 55, 55, 1, 1, 10, 10, 38, 38, 73, 73, 88, 88, 89, 1, 1, 11, 11, 47, 47, 103, 103, 138, 138, 144, 144, 1, 1, 12, 12, 57, 57, 141, 141, 211, 211, 232, 232, 233

Offset

0,6

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Formula

Number triangle T(n, k) = Sum_{j=0..floor(k/2)} binomial(n-j, j) if k <= n, 0 otherwise.

T(n, n) = A000045(n+1).

Sum_{k=0..floor(n/2)} T(n-k, k) = A099574(n).

Sum_{k=0..n} T(n, k) = A029907(n+1).

Antidiagonals of the following array: the first row equals the Fibonacci numbers, (1, 1, 2, 3, 5, ...), and the (n+1)-st row is obtained by the matrix-vector product A128174 * n-th row. - Gary W. Adamson, Jan 19 2011

From G. C. Greubel, Jul 25 2022: (Start)

T(n, n-1) = A052952(n-1), n >= 1.

T(n, n-2) = A054451(n-2), n >= 2.

T(n, n-3) = A099571(n-3), n >= 3.

T(n, n-4) = A099572(n-4), n >= 4. (End)

Example

First few rows of the array:
  1, 1, 2, 3,  5,  8, ... (A000045)
  1, 1, 3, 4,  8, 12, ... (A052952)
  1, 1, 4, 5, 12, 17, ... (A054451)
  1, 1, 5, 6, 17, 23, ... (A099571)
  1, 1, 6, 7, 23, 30, ... (A099572)
  ...
Triangle begins as:
  1;
  1, 1;
  1, 1, 2;
  1, 1, 3, 3;
  1, 1, 4, 4,  5;
  1, 1, 5, 5,  8,  8;
  1, 1, 6, 6, 12, 12, 13;
  1, 1, 7, 7, 17, 17, 21, 21;
  1, 1, 8, 8, 23, 23, 33, 33, 34;
  1, 1, 9, 9, 30, 30, 50, 50, 55, 55;

Mathematica

T[n_, k_]:= Sum[Binomial[n-j, j], {j, 0, Floor[k/2]}];
Table[T[n, k], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Jul 25 2022 *)

Prog

(Magma) [(&+[Binomial(n-j, j): j in [0..Floor(k/2)]]): k in [0..n], n in [0..15]]; // G. C. Greubel, Jul 25 2022
(SageMath)
def A099573(n, k): return sum(binomial(n-j, j) for j in (0..(k//2)))
flatten([[A099573(n, k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Jul 25 2022

Crossrefs

Cf. A000045, A029907 (row sums), A052951, A052952, A054450, A054451, A052952.

Cf. A099571, A099572, A099574 (diagonal sums), A099575.

Sequence in context: A096589 A176427 A324592 * A107430 A330885 A355146

Adjacent sequences: A099570 A099571 A099572 * A099574 A099575 A099576

Keyword

easy,nonn,tabl

Author

Paul Barry, Oct 23 2004

Status

approved