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A122076 Coefficients of a generalized Jaco-Lucas polynomial (even indices) read by rows. 2
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Row sums give A052539. - 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.

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 A170842

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

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Last modified August 19 04:18 EDT 2019. Contains 326109 sequences. (Running on oeis4.)