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A099573 Reverse of number triangle A054450. 3
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 (list; table; graph; refs; listen; history; text; internal format)
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,changed

AUTHOR

Paul Barry, Oct 23 2004

STATUS

approved

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Last modified August 17 08:02 EDT 2022. Contains 356184 sequences. (Running on oeis4.)