A172101 Triangle, read by rows, given by [0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, ...] DELTA [1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, ...] where DELTA is the operator defined in A084938.

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 2, 1, 0, 1, 2, 4, 2, 1, 0, 1, 3, 6, 6, 3, 1, 0, 1, 3, 9, 9, 9, 3, 1, 0, 1, 4, 12, 18, 18, 12, 4, 1, 0, 1, 4, 16, 24, 36, 24, 16, 4, 1, 0, 1, 5, 20, 40, 60, 60, 40, 20, 5, 1, 0, 1, 5, 25, 50, 100, 100, 100, 50, 25, 5, 1, 0, 1, 6, 30, 75, 150, 200, 200, 150, 75, 30, 6, 1

Offset: 0

Author

Philippe Deléham, Jan 25 2010

Keywords

• nonn
• tabl

Comments

Number of symmetric Dyck paths of semilength n with k peaks.

Examples

			Triangle begins :
  1;
  0,  1;
  0,  1,  1;
  0,  1,  1,  1;
  0,  1,  2,  2,  1;
  0,  1,  2,  4,  2,   1;
  0,  1,  3,  6,  6,   3,   1;
  0,  1,  3,  9,  9,   9,   3,   1;
  0,  1,  4, 12, 18,  18,  12,   4,   1;
  0,  1,  4, 16, 24,  36,  24,  16,   4,  1;
  0,  1,  5, 20, 40,  60,  60,  40,  20,  5,  1;
  0,  1,  5, 25, 50, 100, 100, 100,  50, 25,  5,  1;
  0,  1,  6, 30, 75, 150, 200, 200, 150, 75, 30,  6,  1;
		

Links

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

Crossrefs

Cf. A001405 (row sums), A005566, A084938, A088518 (diagonal sums), A088855.

Column k: A000007 (k=0), A000012 (k=1), A008619 (k=2), A002620 (k=3), A028724 (k=4), A028723 (k=5), A028725 (k=6), A331574 (k=7).

Programs

Magma

[n eq 0 select 1 else (&*[Binomial(Floor((n-j)/2), Floor((k-j)/2)): j in [0..1]]): k in [0..n], n in [0..15]]; // G. C. Greubel, Apr 08 2022

Mathematica

T[n_, k_]:= Product[Binomial[Floor[(n-j)/2], Floor[(k-j)/2]], {j,0,1}];
Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 08 2022 *)

Sage

def A172101(n,k):
    if (n==0): return 1
    else: return product(binomial( (n-j)//2, (k-j)//2 ) for j in (0..1))
flatten([[A172101(n,k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Apr 08 2022

Formula

Sum_{k=0..n} T(n,k) = A001405(n).

Sum_{k=0..floor(n/2)} T(n-k, k) = [n=0] - [n=1] + A088518(n)*[n >= 1].

From G. C. Greubel, Apr 08 2022: (Start)

T(n, k) = binomial(floor((n-1)/2), floor((k-1)/2))*binomial(floor(n/2), floor(k/2)).

T(2*n, n) = [n=0] + A005566(n-1)*[n >= 1].

T(n-1, n-k) = T(n-1, k), n >= 1, 1 <= k <= n. (End)