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

1,2

Links

Table of n, a(n) for n=1..55.

Formula

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

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

Example

[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.

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 *)

Prog

(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)]

Crossrefs

Row sums are A002054.

Cf. A001700, A009766.

Sequence in context: A161136 A258163 A108381 * A261469 A292498 A108692

Adjacent sequences: A238758 A238759 A238760 * A238762 A238763 A238764

Keyword

nonn,tabl

Author

Peter Luschny, Mar 05 2014

Status

approved