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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
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.
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
KEYWORD
nonn,tabl
AUTHOR
Philippe Deléham, Jan 25 2010
STATUS
approved