A238761 Subtriangle of the generalized ballot numbers, T(n,k) = A238762(2*k-1,2*n-1), 1<=k<=n, read by rows.

1, 2, 3, 3, 8, 10, 4, 15, 30, 35, 5, 24, 63, 112, 126, 6, 35, 112, 252, 420, 462, 7, 48, 180, 480, 990, 1584, 1716, 8, 63, 270, 825, 1980, 3861, 6006, 6435, 9, 80, 385, 1320, 3575, 8008, 15015, 22880, 24310, 10, 99, 528, 2002, 6006, 15015, 32032, 58344, 87516, 92378

Offset: 1

Author

Peter Luschny, Mar 05 2014

Keywords

• nonn
• tabl

Examples

			[n\k 1   2    3    4    5    6     7 ]
[1]  1,
[2]  2,  3,
[3]  3,  8,  10,
[4]  4, 15,  30,  35,
[5]  5, 24,  63, 112, 126,
[6]  6, 35, 112, 252, 420,  462,
[7]  7, 48, 180, 480, 990, 1584, 1716.
		

Crossrefs

Row sums are A002054.

Cf. A001700, A009766.

Programs

Maple

binom2 := proc(n, k) local h;
   h := n -> (n-((1-(-1)^n)/2))/2;
   n!/(h(n-k)!*h(n+k)!) end:
A238761 := (n, k) -> binom2(n+k, n-k+1)*(n-k+1)/(n+k):
seq(print(seq(A238761(n, k), k=1..n)), n=1..7);

Mathematica

h[n_] := (n - ((1 - (-1)^n)/2))/2;
binom2[n_, k_] := n!/(h[n-k]! h[n+k]!);
T[n_, k_] := binom2[n+k, n-k+1] (n-k+1)/(n+k);
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 10 2019, from Maple *)

Sage

@CachedFunction
def ballot(p, q):
    if p == 0 and q == 0: return 1
    if p < 0 or p > q: return 0
    S = ballot(p-2, q) + ballot(p, q-2)
    if q % 2 == 1: S += ballot(p-1, q-1)
    return S
A238761 = lambda n, k: ballot(2*k-1, 2*n-1)
for n in (1..7): [A238761(n, k) for k in (1..n)]

Formula

T(n,n) = A001700(n-1).

T(n,n-1) = A162551(n-1).