A122076 Coefficients of a generalized Jaco-Lucas polynomial (even indices) read by rows.
2, 3, 2, 7, 8, 2, 18, 30, 15, 2, 47, 104, 80, 24, 2, 123, 340, 355, 170, 35, 2, 322, 1068, 1410, 932, 315, 48, 2, 843, 3262, 5208, 4396, 2079, 532, 63, 2, 2207, 9760, 18280, 18784, 11440, 4144, 840, 80, 2, 5778, 28746, 61785, 74838, 55809, 26226, 7602, 1260
Offset: 0
Examples
The triangle T(n,k) begins: n\k: 0 1 2 3 4 5 6 7 8 9 10 0: 2 1: 3 2 2: 7 8 2 3: 18 30 15 2 4: 47 104 80 24 2 5: 123 340 355 170 35 2 6: 322 1068 1410 932 315 48 2 7: 843 3262 5208 4396 2079 532 63 2 8: 2207 9760 18280 18784 11440 4144 840 80 2 9: 5778 28746 61785 74838 55809 26226 7602 1260 99 2 10: 15127 83620 202840 282980 249815 144488 54690 13080 1815 120 2 ... reformatted and extended. - _Franck Maminirina Ramaharo_, Jul 09 2018
Links
- Muniru A Asiru, Table of n, a(n) for n = 0..1325 (offset adapted by _Georg Fischer_, Jan 31 2019).
- Franck Ramaharo, A one-variable bracket polynomial for some Turk's head knots, arXiv:1807.05256 [math.CO], 2018.
- Yidong Sun, Numerical Triangles and Several Classical Sequences, Fib. Quart. 43, no. 4, (2005) 359-370, Table 3.3.
Crossrefs
Cf. A200073.
Programs
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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
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Mathematica
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 *) -
PARI
t(n,k)={if(n>=1, sum(j=0,n/2, n*binomial(n-j,j)*binomial(j,k)/(n-j)), 2 );} T(n,k) = t(2*n, k); { nmax=10 ; for(n=0,nmax, for(k=0,n, print1(T(n,k),",") ; ); ); }
Formula
T(n,k) = Sum_(j=0..n) 2n*binomial(2n-j,j)*binomial(j,k)/(2n-j).
From Franck Maminirina Ramaharo, Jul 09 2018: (Start)
T(n,0) = A005248(n).
T(n,1) = A099920(2*n-1).
T(n,n-1) = A005563(n).
(End)
Extensions
Offset changed from 1 to 0 by Franck Maminirina Ramaharo, Jul 30 2018
Comments