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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.
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%I #19 Nov 17 2023 12:22:18

%S 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,

%T 95,330,685,873,685,330,95,15,1,1,18,141,630,1770,3258,3989,3258,1770,

%U 630,141,18,1,1,21,196,1071,3801,9198,15533,18483,15533,9198,3801,1071,196,21,1

%N 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.

%C From _R. J. Mathar_, Nov 05 2021: (Start)

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

%C 1 0 1 0 1 0 1 0 1 0 1 ...

%C 0 3 0 6 0 9 0 12 0 15 0 ...

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

%C 0 6 0 45 0 144 0 330 0 630 0 ...

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

%C 0 9 0 144 0 873 0 3258 0 9198 0 ...

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

%C 0 12 0 330 0 3258 0 18483 0 74280 0 ...

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

%C 0 15 0 630 0 9198 0 74280 0 408105 0 ...

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

%H Michael De Vlieger, <a href="/A272866/b272866.txt">Table of n, a(n) for n = 0..10200</a> (rows 0 <= n <= 100, flattened).

%H Feryal Alayont and Evan Henning, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Alayont/ala4.html">Edge Covers of Caterpillars, Cycles with Pendants, and Spider Graphs</a>, J. Int. Seq. (2023) Vol. 26, Art. 23.9.4.

%H László Németh, <a href="http://math.colgate.edu/~integers/t41/t41.Abstract.html">Tetrahedron trinomial coefficient transform</a>, Integers (2019) 19, Article A41.

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

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

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

%e 1;

%e 1, 3, 1;

%e 1, 6, 11, 6, 1;

%e 1, 9, 30, 45, 30, 9, 1;

%e 1, 12, 58, 144, 195, 144, 58, 12, 1;

%e 1, 15, 95, 330, 685, 873, 685, 330, 95, 15, 1;

%p T := (n,k) -> simplify(GegenbauerC(`if`(k<n,k,2*n-k),-n, -3/2)):

%p for n from 0 to 6 do seq(T(n,k),k=0..2*n) od;

%t 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 *)

%Y Cf. A002212, A026375, A026376, A110165.

%K nonn,tabf

%O 0,3

%A _Peter Luschny_, May 08 2016