A370233 Triangle read by rows. T(n, k) = (n - k + 1) * binomial(n + k + 1, 2*k)^2 / (n + k + 1).
1, 1, 3, 1, 18, 5, 1, 60, 75, 7, 1, 150, 525, 196, 9, 1, 315, 2450, 2352, 405, 11, 1, 588, 8820, 17640, 7425, 726, 13, 1, 1008, 26460, 97020, 81675, 18876, 1183, 15, 1, 1620, 69300, 426888, 637065, 286286, 41405, 1800, 17, 1, 2475, 163350, 1585584, 3864861, 3006003, 828100, 81600, 2601, 19
Offset: 0
Examples
Triangle starts: [0] 1; [1] 1, 3; [2] 1, 18, 5; [3] 1, 60, 75, 7; [4] 1, 150, 525, 196, 9; [5] 1, 315, 2450, 2352, 405, 11; [6] 1, 588, 8820, 17640, 7425, 726, 13; [7] 1, 1008, 26460, 97020, 81675, 18876, 1183, 15; [8] 1, 1620, 69300, 426888, 637065, 286286, 41405, 1800, 17;
Programs
-
Maple
T := (n, k) -> (n - k + 1)*binomial(n + k + 1, 2*k)^2/(n + k + 1): seq(print(seq(T(n, k), k = 0..n)), n = 0..8);
-
Mathematica
P[n_, z_] := HypergeometricPFQ[{-1 - n, -n, 1 + n, 2 + n}, {1/2, 1/2, 1}, z/16]; Table[CoefficientList[P[n, z], z], {n, 0, 9}] // Flatten
Formula
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.
T(n, k) = [z^k] hypergeom([-1 - n, -n, 1 + n, 2 + n], [1/2, 1/2, 1], z/16).