This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A371395 #31 Mar 29 2025 20:07:06
%S A371395 1,1,1,2,3,2,5,10,10,5,14,35,45,35,14,42,126,196,196,126,42,132,462,
%T A371395 840,1008,840,462,132,429,1716,3564,4950,4950,3564,1716,429,1430,6435,
%U A371395 15015,23595,27225,23595,15015,6435,1430
%N A371395 Triangle read by rows: T(n, k) = binomial(n + k, k) * binomial(2*n - k, n - k) / (n + 1).
%C A371395 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 A371395 The triangle is symmetric under the exchange of k with n - k.
%F A371395 From _Peter Luschny_, Mar 21 2024: (Start)
%F A371395 T(n, k) = hypergeom([-n, -k], [1], 1)*hypergeom([-n, k - n], [1], 1)/(n + 1).
%F A371395 2^n*Sum_{k=0..n} T(n, k)*(1/2)^k = A085614(n + 1).
%F A371395 2^n*Sum_{k=0..n} T(n, k)*(-1/2)^k = A250886(n + 1). (End)
%e A371395 Triangle begins:
%e A371395 [0] [ 1],
%e A371395 [1] [ 1, 1],
%e A371395 [2] [ 2, 3, 2],
%e A371395 [3] [ 5, 10, 10, 5],
%e A371395 [4] [14, 35, 45, 35, 14],
%e A371395 [5] [42, 126, 196, 196, 126, 42].
%p A371395 T := (n, k) -> binomial(n + k, k)*binomial(2*n - k, n)/(n + 1):
%p A371395 seq(print(seq(T(n, k), k = 0..n)), n = 0..7); # _Peter Luschny_, Mar 21 2024
%t A371395 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 A371395 (* _Peter Luschny_, Mar 21 2024 *)
%o A371395 (SageMath)
%o A371395 def Trow(n):
%o A371395 return [binomial(n+k, k) * binomial(2*n-k, n-k) / (n+1) for k in range(n+1)]
%o A371395 (SageMath) # As the reverse of x*(1-x)*(1-t*x) w.r.t variable x.
%o A371395 t = polygen(QQ, 't')
%o A371395 x = LazyPowerSeriesRing(t.parent(), 'x').0
%o A371395 gf = x*(1-x)*(1-t*x)
%o A371395 coeffs = gf.revert() / x
%o A371395 for n in range(6):
%o A371395 print(coeffs[n].list())
%Y A371395 Column 0 and main diagonal are A000108.
%Y A371395 Column 1 and subdiagonal are A001700.
%Y A371395 Row sums are A006013.
%Y A371395 The even bisection of the alternating row sums is A001764.
%Y A371395 The central terms are A188681.
%Y A371395 Cf. A085614, A250886, A371400.
%K A371395 nonn,tabl,easy
%O A371395 0,4
%A A371395 _F. Chapoton_, Mar 21 2024