A272866 Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k=0 and 0<=k<=2n.

1, 1, 3, 1, 1, 6, 11, 6, 1, 1, 9, 30, 45, 30, 9, 1, 1, 12, 58, 144, 195, 144, 58, 12, 1, 1, 15, 95, 330, 685, 873, 685, 330, 95, 15, 1, 1, 18, 141, 630, 1770, 3258, 3989, 3258, 1770, 630, 141, 18, 1, 1, 21, 196, 1071, 3801, 9198, 15533, 18483, 15533, 9198, 3801, 1071, 196, 21, 1

Offset: 0

Author

Peter Luschny, May 08 2016

Keywords

• nonn
• tabf

Comments

From R. J. Mathar, Nov 05 2021: (Start)

These are the antidiagonals of the following array with the bivariate generating function 1/(1-x^2-3*x*y-y^2):

1 0 1 0 1 0 1 0 1 0 1 ...

0 3 0 6 0 9 0 12 0 15 0 ...

1 0 11 0 30 0 58 0 95 0 141 ...

0 6 0 45 0 144 0 330 0 630 0 ...

1 0 30 0 195 0 685 0 1770 0 3801 ...

0 9 0 144 0 873 0 3258 0 9198 0 ...

1 0 58 0 685 0 3989 0 15533 0 46928 ...

0 12 0 330 0 3258 0 18483 0 74280 0 ...

1 0 95 0 1770 0 15533 0 86515 0 356283 ...

0 15 0 630 0 9198 0 74280 0 408105 0 ...

1 0 141 0 3801 0 46928 0 356283 0 1936881 ... (End)

Examples

			                                1;
                            1,  3, 1;
                         1, 6, 11, 6, 1;
                     1, 9, 30, 45, 30, 9, 1;
              1, 12, 58, 144, 195, 144, 58, 12, 1;
         1, 15, 95, 330, 685, 873, 685, 330, 95, 15, 1;
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..10200 (rows 0 <= n <= 100, flattened).
• Feryal Alayont and Evan Henning, Edge Covers of Caterpillars, Cycles with Pendants, and Spider Graphs, J. Int. Seq. (2023) Vol. 26, Art. 23.9.4.
• Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 6.
• László Németh, Tetrahedron trinomial coefficient transform, Integers (2019) 19, Article A41.

Crossrefs

Cf. A002212, A026375, A026376, A110165.

Programs

Maple

T := (n,k) -> simplify(GegenbauerC(`if`(k
				

Mathematica

Table[If[n == 0, 1, GegenbauerC[If[k < n, k, 2 n - k], -n, -3/2]], {n, 0, 7}, {k, 0, 2 n}] // Flatten (* Michael De Vlieger, Aug 02 2019 *)

Formula

T(n,n) = A026375(n) for n>=0.

T(n,n-1) = A026376(n) for n>=1.

T(n,n+1)/n = A002212(n) for n>=1.