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 A370233 #10 Feb 14 2024 05:44:01
%S A370233 1,1,3,1,18,5,1,60,75,7,1,150,525,196,9,1,315,2450,2352,405,11,1,588,
%T A370233 8820,17640,7425,726,13,1,1008,26460,97020,81675,18876,1183,15,1,1620,
%U A370233 69300,426888,637065,286286,41405,1800,17,1,2475,163350,1585584,3864861,3006003,828100,81600,2601,19
%N A370233 Triangle read by rows. T(n, k) = (n - k + 1) * binomial(n + k + 1, 2*k)^2 / (n + k + 1).
%F A370233 T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} binomial(n + k, 2*k) * Pochhammer(n - k + c, 2*k) * z^k / (2*k)! and c = 2.
%F A370233 T(n, k) = [z^k] hypergeom([-1 - n, -n, 1 + n, 2 + n], [1/2, 1/2, 1], z/16).
%e A370233 Triangle starts:
%e A370233 [0] 1;
%e A370233 [1] 1, 3;
%e A370233 [2] 1, 18, 5;
%e A370233 [3] 1, 60, 75, 7;
%e A370233 [4] 1, 150, 525, 196, 9;
%e A370233 [5] 1, 315, 2450, 2352, 405, 11;
%e A370233 [6] 1, 588, 8820, 17640, 7425, 726, 13;
%e A370233 [7] 1, 1008, 26460, 97020, 81675, 18876, 1183, 15;
%e A370233 [8] 1, 1620, 69300, 426888, 637065, 286286, 41405, 1800, 17;
%p A370233 T := (n, k) -> (n - k + 1)*binomial(n + k + 1, 2*k)^2/(n + k + 1):
%p A370233 seq(print(seq(T(n, k), k = 0..n)), n = 0..8);
%t A370233 P[n_, z_] := HypergeometricPFQ[{-1 - n, -n, 1 + n, 2 + n}, {1/2, 1/2, 1}, z/16];
%t A370233 Table[CoefficientList[P[n, z], z], {n, 0, 9}] // Flatten
%Y A370233 Cf. A370232 (c=1), A370234 (row sums).
%K A370233 nonn,tabl
%O A370233 0,3
%A A370233 _Peter Luschny_, Feb 13 2024