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

0,3

Links

Table of n, a(n) for n=0..54.

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).

Example

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;

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

Crossrefs

Cf. A370232 (c=1), A370234 (row sums).

Sequence in context: A089974 A346039 A143849 * A335689 A105626 A071210

Adjacent sequences: A370230 A370231 A370232 * A370234 A370235 A370236

Keyword

nonn,tabl

Author

Peter Luschny, Feb 13 2024

Status

approved