A182579 Triangle read by rows: T(0,0) = 1, for n>0: T(n,n) = 2 and for k<=floor(n/2): T(n,2*k) = n/(n-k) * binomial(n-k,k), T(n,2*k+1) = (n-1)/(n-1-k) * binomial(n-1-k,k).

1, 1, 2, 1, 1, 2, 1, 1, 3, 2, 1, 1, 4, 3, 2, 1, 1, 5, 4, 5, 2, 1, 1, 6, 5, 9, 5, 2, 1, 1, 7, 6, 14, 9, 7, 2, 1, 1, 8, 7, 20, 14, 16, 7, 2, 1, 1, 9, 8, 27, 20, 30, 16, 9, 2, 1, 1, 10, 9, 35, 27, 50, 30, 25, 9, 2, 1, 1, 11, 10, 44, 35, 77, 50, 55, 25, 11, 2

Offset: 0

Author

Reinhard Zumkeller, May 06 2012

Keywords

• nonn
• tabl

Comments

A000204(n+1) = sum of n-th row, Lucas numbers;

A000204(n+3) = alternating row sum of n-th row;

A182584(n) = T(2*n,n), central terms;

A000012(n) = T(n,0), left edge;

A040000(n) = T(n,n), right edge;

A054977(n-1) = T(n,1) for n > 0;

A109613(n-1) = T(n,n-1) for n > 0;

A008794(n) = T(n,n-2) for n > 1.

Examples

			Starting with 2nd row = [1 2] the rows of the triangle are defined recursively without computing explicitely binomial coefficients; demonstrated for row 8, (see also Haskell program):
   (0) 1  1  7  6 14  9  7  2      [A]  row 7 prepended by 0
    1  1  7  6 14  9  7  2 (0)     [B]  row 7, 0 appended
    1  0  1  0  1  0  1  0  1      [C]  1 and 0 alternating
    1  0  7  0 14  0  7  0  0      [D]  = [B] multiplied by [C]
    1  1  8  7 20 14 16  7  2      [E]  = [D] added to [A] = row 8.
The triangle begins:                 | A000204
              1                      |       1
             1  2                    |       3
            1  1  2                  |       4
           1  1  3  2                |       7
          1  1  4  3  2              |      11
         1  1  5  4  5  2            |      18
        1  1  6  5  9  5  2          |      29
       1  1  7  6 14  9  7  2        |      47
      1  1  8  7 20 14 16  7  2      |      76
     1  1  9  8 27 20 30 16  9  2    |     123
    1  1 10  9 35 27 50 30 25  9  2  |     199 .
		

Links

• Reinhard Zumkeller, Rows n = 0..150 of triangle, flattened
• Henry W. Gould, A Variant of Pascal's Triangle, The Fibonacci Quarterly, Vol. 3, Nr. 4, Dec. 1965, pp. 257-271, with corrections.

Crossrefs

Cf. A065941, A059841.

Programs

Haskell

a182579 n k = a182579_tabl !! n !! k
a182579_row n = a182579_tabl !! n
a182579_tabl = [1] : iterate (\row ->
  zipWith (+) ([0] ++ row) (zipWith (*) (row ++ [0]) a059841_list)) [1,2]

Mathematica

T[_, 0] = 1;
T[n_, n_] /; n > 0 = 2;
T[_, 1] = 1;
T[n_, k_] := T[n, k] = Which[
     OddQ[k],  T[n - 1, k - 1],
     EvenQ[k], T[n - 1, k - 1] + T[n - 1, k]];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 01 2021 *)

Formula

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