A371395 - OEIS

%I #26 Mar 22 2024 15:12:37

%S 1,1,1,2,3,2,5,10,10,5,14,35,45,35,14,42,126,196,196,126,42,132,462,

%T 840,1008,840,462,132,429,1716,3564,4950,4950,3564,1716,429,1430,6435,

%U 15015,23595,27225,23595,15015,6435,1430

%N Triangle read by rows: T(n, k) = binomial(n + k, k) * binomial(2*n - k, n - k) / (n + 1).

%C 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).

%C The triangle is symmetric under the exchange of k with n - k.

%F From _Peter Luschny_, Mar 21 2024: (Start)

%F T(n, k) = (hypergeom([-n, -k], [1], 1]*hypergeom([-n, k - n], [1], 1])/(n + 1).

%F 2^n*Sum_{k=0..n} T(n, k)*(1/2)^k = A085614(n + 1).

%F 2^n*Sum_{k=0..n} T(n, k)*(-1/2)^k = A250886(n + 1).  (End)

%e Triangle begins:

%e   [0] [ 1],

%e   [1] [ 1,   1],

%e   [2] [ 2,   3,   2],

%e   [3] [ 5,  10,  10,   5],

%e   [4] [14,  35,  45,  35,  14],

%e   [5] [42, 126, 196, 196, 126, 42].

%p T := (n, k) -> binomial(n + k, k)*binomial(2*n - k, n)/(n + 1):

%p seq(print(seq(T(n, k), k = 0..n)), n = 0..7);  # _Peter Luschny_, Mar 21 2024

%t 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

%t (* _Peter Luschny_, Mar 21 2024 *)

%o (Sage)

%o def Trow(n):

%o     return [binomial(n+k, k) * binomial(2*n-k, n-k) / (n+1) for k in range(n+1)

%o (Sage)  # As the reverse of x*(1-x)*(1-t*x) w.r.t variable x.

%o t = polygen(QQ, 't')

%o x = LazyPowerSeriesRing(t.parent(), 'x').0

%o gf = x*(1-x)*(1-t*x)

%o coeffs = gf.revert() / x

%o for n in range(6):

%o     print(coeffs[n].list())

%Y Column 0 and main diagonal are A000108.

%Y Column 1 and subdiagonal are A001700.

%Y Row sums are A006013.

%Y The even bisection of the alternating row sums is A001764.

%Y The central terms are A188681.

%Y Cf. A085614, A250886, A371400.

%K nonn,tabl,easy

%O 0,4

%A _F. Chapoton_, Mar 21 2024