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
KEYWORD
AUTHOR
R. J. Mathar, Oct 16 2006
EXTENSIONS
Offset changed from 1 to 0 by Franck Maminirina Ramaharo, Jul 30 2018
STATUS
approved