|
%I #10 Apr 08 2022 06:57:24
%S 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,
%T 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,
%U 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
%N 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.
%C Number of symmetric Dyck paths of semilength n with k peaks.
%H G. C. Greubel, <a href="/A172101/b172101.txt">Rows n = 0..50 of the triangle, flattened</a>
%F Sum_{k=0..n} T(n,k) = A001405(n).
%F Sum_{k=0..floor(n/2)} T(n-k, k) = [n=0] - [n=1] + A088518(n)*[n >= 1].
%F From _G. C. Greubel_, Apr 08 2022: (Start)
%F T(n, k) = binomial(floor((n-1)/2), floor((k-1)/2))*binomial(floor(n/2), floor(k/2)).
%F T(2*n, n) = [n=0] + A005566(n-1)*[n >= 1].
%F T(n-1, n-k) = T(n-1, k), n >= 1, 1 <= k <= n. (End)
%e Triangle begins :
%e 1;
%e 0, 1;
%e 0, 1, 1;
%e 0, 1, 1, 1;
%e 0, 1, 2, 2, 1;
%e 0, 1, 2, 4, 2, 1;
%e 0, 1, 3, 6, 6, 3, 1;
%e 0, 1, 3, 9, 9, 9, 3, 1;
%e 0, 1, 4, 12, 18, 18, 12, 4, 1;
%e 0, 1, 4, 16, 24, 36, 24, 16, 4, 1;
%e 0, 1, 5, 20, 40, 60, 60, 40, 20, 5, 1;
%e 0, 1, 5, 25, 50, 100, 100, 100, 50, 25, 5, 1;
%e 0, 1, 6, 30, 75, 150, 200, 200, 150, 75, 30, 6, 1;
%t T[n_, k_]:= Product[Binomial[Floor[(n-j)/2], Floor[(k-j)/2]], {j,0,1}];
%t Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* _G. C. Greubel_, Apr 08 2022 *)
%o (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
%o (Sage)
%o def A172101(n,k):
%o if (n==0): return 1
%o else: return product(binomial( (n-j)//2, (k-j)//2 ) for j in (0..1))
%o flatten([[A172101(n,k) for k in (0..n)] for n in (0..15)]) # _G. C. Greubel_, Apr 08 2022
%Y Cf. A001405 (row sums), A005566, A084938, A088518 (diagonal sums), A088855.
%Y 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).
%K nonn,tabl
%O 0,13
%A _Philippe Deléham_, Jan 25 2010
|
