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

0,13

Comments

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

Links

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

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)

Example

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;

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 *)

Prog

(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
(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

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).

Sequence in context: A039800 A309416 A251418 * A301365 A109246 A037890

Adjacent sequences: A172098 A172099 A172100 * A172102 A172103 A172104

Keyword

nonn,tabl

Author

Philippe Deléham, Jan 25 2010

Status

approved