A156580 Triangle T(n, k) = binomial(n-1, k-1) for n < 4, 2*(n + k - 4) + (-1)^n if k = (n+1)/2, 2*(n + k - 4) if k < floor(n/2) + 1, otherwise 2*(2*n - k - 3), with T(n, 0) = T(n, n) = 1, read by rows.

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 6, 7, 6, 1, 1, 8, 10, 10, 8, 1, 1, 10, 12, 13, 12, 10, 1, 1, 12, 14, 16, 16, 14, 12, 1, 1, 14, 16, 18, 19, 18, 16, 14, 1, 1, 16, 18, 20, 22, 22, 20, 18, 16, 1, 1, 18, 20, 22, 24, 25, 24, 22, 20, 18, 1, 1, 20, 22, 24, 26, 28, 28, 26, 24, 22, 20, 1

Offset

1,5

Links

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Formula

T(n, k) = binomial(n-1, k-1) for n < 4, 2*(n + k - 4) + (-1)^n if k = (n+1)/2, 2*(n + k - 4) if k < floor(n/2) + 1, otherwise 2*(2*n - k - 3), with T(n, 0) = T(n, n) = 1.

T(2*n, k) = T(2*n, 2*n-k+1).

Example

Triangle begins as:
  1;
  1,  1;
  1,  2,  1;
  1,  4,  4,  1;
  1,  6,  7,  6,  1;
  1,  8, 10, 10,  8,  1;
  1, 10, 12, 13, 12, 10,  1;
  1, 12, 14, 16, 16, 14, 12,  1;
  1, 14, 16, 18, 19, 18, 16, 14,  1;
  1, 16, 18, 20, 22, 22, 20, 18, 16,  1;
  1, 18, 20, 22, 24, 25, 24, 22, 20, 18,  1;
  1, 20, 22, 24, 26, 28, 28, 26, 24, 22, 20, 1;

Mathematica

T[n_, k_]:= T[n, k]= If[k==1 || k==n, 1, If[n<4, Binomial[n-1, k-1], If[k==(n+1)/2, 2*(n+k-4) + (-1)^n, If[Floor[n/2]>(k-1), 2*(n+k-4), 2*(2*n-k-3) ]]]];
Table[T[n, k], {n, 15}, {k, n}]//Flatten (* modified by G. C. Greubel, Jan 04 2022 *)

Prog

(Sage)
def T(n, k):
    if (k==1 or k==n): return 1
    elif (n<4): return binomial(n-1, k-1)
    elif (k==(n+1)/2): return 2*(n+k-4) + (-1)^n
    elif (k<(n//2)+1): return 2*(n+k-4)
    else: return 2*(2*n - k - 3)
[[T(n, k) for k in (1..n)] for n in (1..15)] # G. C. Greubel, Jan 04 2022

Crossrefs

Sequence in context: A026386 A147532 A283796 * A157528 A132731 A128966

Adjacent sequences: A156577 A156578 A156579 * A156581 A156582 A156583

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 10 2009

Extensions

Edited by G. C. Greubel, Jan 04 2022

Status

approved