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A122076 Coefficients of a generalized Jaco-Lucas polynomial (even indices) read by rows. 2

%I

%S 2,3,2,7,8,2,18,30,15,2,47,104,80,24,2,123,340,355,170,35,2,322,1068,

%T 1410,932,315,48,2,843,3262,5208,4396,2079,532,63,2,2207,9760,18280,

%U 18784,11440,4144,840,80,2,5778,28746,61785,74838,55809,26226,7602,1260

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

%C Row sums give A052539. - _Franck Maminirina Ramaharo_, Jul 09 2018

%H 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 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 Yidong Sun, <a href="http://www.combinatorics.cn/publications/papers/2004/Triangle.pdf">Numerical Triangles and Several Classical Sequences</a>, Fib. Quart. 43, no. 4, (2005) 359-370, Table 3.3.

%F T(n,k) = Sum_(j=0..n) 2n*binomial(2n-j,j)*binomial(j,k)/(2n-j).

%F From _Franck Maminirina Ramaharo_, Jul 09 2018: (Start)

%F T(n,0) = A005248(n).

%F T(n,1) = A099920(2*n-1).

%F T(n,n-1) = A005563(n).

%F (End)

%e The triangle T(n,k) begins:

%e n\k: 0 1 2 3 4 5 6 7 8 9 10

%e 0: 2

%e 1: 3 2

%e 2: 7 8 2

%e 3: 18 30 15 2

%e 4: 47 104 80 24 2

%e 5: 123 340 355 170 35 2

%e 6: 322 1068 1410 932 315 48 2

%e 7: 843 3262 5208 4396 2079 532 63 2

%e 8: 2207 9760 18280 18784 11440 4144 840 80 2

%e 9: 5778 28746 61785 74838 55809 26226 7602 1260 99 2

%e 10: 15127 83620 202840 282980 249815 144488 54690 13080 1815 120 2

%e ... reformatted and extended. - _Franck Maminirina Ramaharo_, Jul 09 2018

%t 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 (PARI) t(n,k)={if(n>=1, sum(j=0,n/2, n*binomial(n-j,j)*binomial(j,k)/(n-j)), 2 );}

%o T(n,k) = t(2*n, k);

%o { nmax=10 ; for(n=0,nmax, for(k=0,n, print1(T(n,k),",") ; ); ); }

%o (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 Cf. A200073.

%K easy,nonn,tabl

%O 0,1

%A _R. J. Mathar_, Oct 16 2006

%E Offset changed from 1 to 0 by _Franck Maminirina Ramaharo_, Jul 30 2018

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Last modified September 17 16:57 EDT 2019. Contains 327136 sequences. (Running on oeis4.)