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

0,1

Comments

Row sums give A052539. - Franck Maminirina Ramaharo, Jul 09 2018

Alternating row sums seem to be 1, except when n=0. - F. Chapoton, Nov 09 2021

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.

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)

Example

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

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 *)

Prog

(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), ", ") ; ); ); }
(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

Crossrefs

Cf. A200073.

Sequence in context: A129022 A210564 A208930 * A209774 A271322 A373420

Adjacent sequences: A122073 A122074 A122075 * A122077 A122078 A122079

Keyword

easy,nonn,tabl

Author

R. J. Mathar, Oct 16 2006

Extensions

Offset changed from 1 to 0 by Franck Maminirina Ramaharo, Jul 30 2018

Status

approved