A272866 Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=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

0,3

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)

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.

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.

Example

                                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;

Maple

T := (n, k) -> simplify(GegenbauerC(`if`(k<n, k, 2*n-k), -n, -3/2)):
for n from 0 to 6 do seq(T(n, k), k=0..2*n) od;

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 *)

Crossrefs

Cf. A002212, A026375, A026376, A110165.

Sequence in context: A235113 A235116 A235114 * A228899 A255918 A102479

Adjacent sequences: A272863 A272864 A272865 * A272867 A272868 A272869

Keyword

nonn,tabf

Author

Peter Luschny, May 08 2016

Status

approved