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A122076 Coefficients of a generalized Jaco-Lucas polynomial (even indices) read by rows.

%I A122076 #48 Jan 05 2025 19:51:38
%S A122076 2,3,2,7,8,2,18,30,15,2,47,104,80,24,2,123,340,355,170,35,2,322,1068,
%T A122076 1410,932,315,48,2,843,3262,5208,4396,2079,532,63,2,2207,9760,18280,
%U A122076 18784,11440,4144,840,80,2,5778,28746,61785,74838,55809,26226,7602,1260
%N A122076 Coefficients of a generalized Jaco-Lucas polynomial (even indices) read by rows.
%C A122076 Row sums give A052539. - _Franck Maminirina Ramaharo_, Jul 09 2018
%C A122076 Alternating row sums seem to be 1, except when n=0. - _F. Chapoton_, Nov 09 2021
%H A122076 Muniru A Asiru, <a href="/A122076/b122076.txt">Table of n, a(n) for n = 0..1325</a> (offset adapted by _Georg Fischer_, Jan 31 2019).
%H A122076 Franck Ramaharo, <a href="https://arxiv.org/abs/1807.05256">A one-variable bracket polynomial for some Turk's head knots</a>, arXiv:1807.05256 [math.CO], 2018.
%H A122076 Yidong Sun, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/43-4/paper43-4-10b.pdf">Numerical Triangles and Several Classical Sequences</a>, Fib. Quart. 43, no. 4, (2005) 359-370, Table 3.3.
%F A122076 T(n,k) = Sum_(j=0..n) 2n*binomial(2n-j,j)*binomial(j,k)/(2n-j).
%F A122076 From _Franck Maminirina Ramaharo_, Jul 09 2018: (Start)
%F A122076 T(n,0) = A005248(n).
%F A122076 T(n,1) = A099920(2*n-1).
%F A122076 T(n,n-1) = A005563(n).
%F A122076 (End)
%e A122076 The triangle T(n,k) begins:
%e A122076 n\k:    0      1       2       3       4       5      6      7     8    9 10
%e A122076 0:      2
%e A122076 1:      3      2
%e A122076 2:      7      8       2
%e A122076 3:     18     30      15       2
%e A122076 4:     47    104      80      24       2
%e A122076 5:    123    340     355     170      35       2
%e A122076 6:    322   1068    1410     932     315      48      2
%e A122076 7:    843   3262    5208    4396    2079     532     63      2
%e A122076 8:   2207   9760   18280   18784   11440    4144    840     80     2
%e A122076 9:   5778  28746   61785   74838   55809   26226   7602   1260    99    2
%e A122076 10: 15127  83620  202840  282980  249815  144488  54690  13080  1815  120  2
%e A122076 ... reformatted and extended. - _Franck Maminirina Ramaharo_, Jul 09 2018
%t A122076 T[n_, k_] := Sum[ 2n*Binomial[2n - j, j]*Binomial[j, k]/(2n - j), {j, 0, n}]; T[0, 0] = 2; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Robert G. Wilson v_, Jul 23 2018 *)
%o A122076 (PARI) t(n,k)={if(n>=1, sum(j=0,n/2, n*binomial(n-j,j)*binomial(j,k)/(n-j)), 2 );}
%o A122076 T(n,k) = t(2*n, k);
%o A122076 { nmax=10 ; for(n=0,nmax, for(k=0,n, print1(T(n,k),",") ; ); ); }
%o A122076 (GAP) Concatenation([2],Flat(List([1..10],n->List([0..n],k->Sum([0..n],j->2*n*Binomial(2*n-j,j)*Binomial(j,k)/(2*n-j)))))); # _Muniru A Asiru_, Jul 27 2018
%Y A122076 Cf. A200073.
%K A122076 easy,nonn,tabl
%O A122076 0,1
%A A122076 _R. J. Mathar_, Oct 16 2006
%E A122076 Offset changed from 1 to 0 by _Franck Maminirina Ramaharo_, Jul 30 2018