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A371395
Triangle read by rows: T(n, k) = binomial(n + k, k) * binomial(2*n - k, n - k) / (n + 1).
1
1, 1, 1, 2, 3, 2, 5, 10, 10, 5, 14, 35, 45, 35, 14, 42, 126, 196, 196, 126, 42, 132, 462, 840, 1008, 840, 462, 132, 429, 1716, 3564, 4950, 4950, 3564, 1716, 429, 1430, 6435, 15015, 23595, 27225, 23595, 15015, 6435, 1430
OFFSET
0,4
COMMENTS
The terms can be seen as graded dimensions of a non-symmetric operad. The Koszul dual operad has Hilbert series x*(1 + x)*(1 + tx). So the current table has as Hilbert series the reverse of x*(1-x)*(1-t*x) w.r.t to x (see Sage below).
The triangle is symmetric under the exchange of k with n - k.
FORMULA
From Peter Luschny, Mar 21 2024: (Start)
T(n, k) = (hypergeom([-n, -k], [1], 1]*hypergeom([-n, k - n], [1], 1])/(n + 1).
2^n*Sum_{k=0..n} T(n, k)*(1/2)^k = A085614(n + 1).
2^n*Sum_{k=0..n} T(n, k)*(-1/2)^k = A250886(n + 1). (End)
EXAMPLE
Triangle begins:
[0] [ 1],
[1] [ 1, 1],
[2] [ 2, 3, 2],
[3] [ 5, 10, 10, 5],
[4] [14, 35, 45, 35, 14],
[5] [42, 126, 196, 196, 126, 42].
MAPLE
T := (n, k) -> binomial(n + k, k)*binomial(2*n - k, n)/(n + 1):
seq(print(seq(T(n, k), k = 0..n)), n = 0..7); # Peter Luschny, Mar 21 2024
MATHEMATICA
T[n_, k_] := (Hypergeometric2F1[-n, -k, 1, 1] Hypergeometric2F1[-n, k - n, 1, 1]) /(n + 1); Table[T[n, k], {n, 0, 7}, {k, 0, n}] // Flatten
(* Peter Luschny, Mar 21 2024 *)
PROG
(Sage)
def Trow(n):
return [binomial(n+k, k) * binomial(2*n-k, n-k) / (n+1) for k in range(n+1)
(Sage) # As the reverse of x*(1-x)*(1-t*x) w.r.t variable x.
t = polygen(QQ, 't')
x = LazyPowerSeriesRing(t.parent(), 'x').0
gf = x*(1-x)*(1-t*x)
coeffs = gf.revert() / x
for n in range(6):
print(coeffs[n].list())
CROSSREFS
Column 0 and main diagonal are A000108.
Column 1 and subdiagonal are A001700.
Row sums are A006013.
The even bisection of the alternating row sums is A001764.
The central terms are A188681.
Sequence in context: A353299 A349790 A335362 * A059098 A082050 A353956
KEYWORD
nonn,tabl,easy
AUTHOR
F. Chapoton, Mar 21 2024
STATUS
approved