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A238761 Subtriangle of the generalized ballot numbers, T(n,k) = A238762(2*k-1,2*n-1), 1<=k<=n, read by rows. 2
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
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.
Sequence in context: A161136 A258163 A108381 * A261469 A292498 A108692
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Mar 05 2014
STATUS
approved

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Last modified July 22 13:05 EDT 2024. Contains 374499 sequences. (Running on oeis4.)