A371395 Triangle read by rows: T(n, k) = binomial(n + k, k) * binomial(2*n - k, n - k) / (n + 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

Author

F. Chapoton, Mar 21 2024

Keywords

• nonn
• tabl
• easy

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.

Examples

			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].
		

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.

Cf. A085614, A250886, A371400.

Programs

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 *)

SageMath

def Trow(n):
    return [binomial(n+k, k) * binomial(2*n-k, n-k) / (n+1) for k in range(n+1)]

SageMath

# 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())

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)