A258708 Triangle read by rows: T(i,j) = integer part of binomial(i+j, i-j)/(2*j+1) for i >= 1 and j = 0..i-1.

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 7, 4, 1, 1, 7, 14, 12, 5, 1, 1, 9, 25, 30, 18, 6, 1, 1, 12, 42, 66, 55, 26, 7, 1, 1, 15, 66, 132, 143, 91, 35, 8, 1, 1, 18, 99, 245, 334, 273, 140, 45, 9, 1, 1, 22, 143, 429, 715, 728, 476, 204, 57, 10, 1

Offset: 1

Author

N. J. A. Sloane, Jun 12 2015

Keywords

• nonn
• tabl

Comments

In the Loh-Shannon-Horadam paper, Table 3 contains a typo (see Extensions lines).

T(n,k) = round(A258993(n,k)/(2*k+1)). - Reinhard Zumkeller, Jun 22 2015

From Reinhard Zumkeller, Jun 23 2015: (Start)

(using tables 4 and 5 of the Loh-Shannon-Horadam paper, p. 8f).

T(n, n-1) = 1;

T(n, n-2) = n for n > 1;

T(n, n-3) = A000969(n-3) for n > 2;

T(n, n-4) = A000330(n-3) for n > 3;

T(n, n-5) = T(2*n-7, 2) = A000970(n) for n > 4;

T(n, n-6) = A000971(n) for n > 5;

T(n, n-7) = A000972(n) for n > 6;

T(n, n-8) = A000973(n) for n > 7;

T(n, 1) = A001840(n-1) for n > 1;

T(2*n, n) = A001764(n);

T(3*n-1, 1) = A000326(n);

T(3*n, 2*n) = A002294(n);

T(4*n, 3*n) = A002296(n). (End)

Examples

			Triangle T(i, j) (with rows i >= 1 and columns j >= 0) begins as follows:
  1;
  1,  1;
  1,  2,  1;
  1,  3,  3,   1;
  1,  5,  7,   4,   1;
  1,  7, 14,  12,   5,   1;
  1,  9, 25,  30,  18,   6,   1;
  1, 12, 42,  66,  55,  26,   7,  1;
  1, 15, 66, 132, 143,  91,  35,  8, 1;
  1, 18, 99, 245, 334, 273, 140, 45, 9, 1;
  ...
		

Links

• Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened
• R. P. Loh, A. G. Shannon, and A. F. Horadam, Divisibility Criteria and Sequence Generators Associated with Fermat Coefficients, Preprint, 1980.

Crossrefs

Cf. A011793.

Cf. A007318, A258993, A158405, A005408, A006013 (central terms).

Cf. A000326, A000330, A000969, A000970, A000971, A000972, A000973, A000976, A001764, A001840, A002294, A002296.

Programs

Haskell

a258708 n k = a258708_tabl !! (n-1) !! k
a258708_row n = a258708_tabl !! (n-1)
a258708_tabl = zipWith (zipWith ((round .) . ((/) `on` fromIntegral)))
                       a258993_tabl a158405_tabl
-- Reinhard Zumkeller, Jun 22 2015, Jun 16 2015

Extensions

Corrected T(8,5) = 26 from Reinhard Zumkeller, Jun 13 2015