OFFSET
0,7
COMMENTS
Sum of rows (see example) gives A225826.
By columns:
T(n,1) is A004526.
T(n,2) is A000217.
T(n,3) is A225972.
T(n,4) is A071239.
T(n,5) is A222715.
T(n,6) is A228581.
T(n,7) is A228582.
T(n,8) is A228583.
Also the number of equivalence classes of ways of placing k 1 X 1 tiles in an n X 2 rectangle under all symmetry operations of the rectangle. - Christopher Hunt Gribble, Feb 16 2014
LINKS
Yosu Yurramendi and María Merino, Rows 0..40 of triangle, flattened
FORMULA
If k even, 4*T(n,k) = binomial(2*n,k) +3*binomial(n,k/2). - Yosu Yurramendi, María Merino, Aug 25 2013
If k odd, 4*T(n,k) = 4*T(n,k) = binomial(2*n,k) +(1-(-1)^n)*binomial(n-1,(k-1)/2). - Yosu Yurramendi, María Merino, Aug 25 2013 [corrected by Christian Barrientos, Jun 14 2018]
EXAMPLE
n\k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
0 1
1 1 1 1
2 1 1 3 1 1
3 1 2 6 6 6 2 1
4 1 2 10 14 22 14 10 2 1
5 1 3 15 32 60 66 60 32 15 3 1
6 1 3 21 55 135 198 246 198 135 55 21 3 1
7 1 4 28 94 266 508 777 868 777 508 266 94 28 4 1
8 1 4 36 140...
...
The length of row n is 2*n+1, so n>= floor((k+1)/2).
MAPLE
A226048 := proc(n, k)
if type(k, 'even') then
binomial(2*n, k) +3*binomial(n, k/2) ;
else
binomial(2*n, k) +(1-(-1)^n)*binomial(n-1, (k-1)/2) ;
end if ;
%/4 ;
end proc:
seq(seq(A226048(n, k), k=0..2*n), n=0..8) ; # R. J. Mathar, Jun 07 2020
MATHEMATICA
T[n_, k_] := If[EvenQ[k],
Binomial[2n, k] + 3 Binomial[n, k/2],
Binomial[2n, k] + (1-(-1)^n) Binomial[n-1, (k-1)/2]]/4;
Table[T[n, k], {n, 0, 8}, { k, 0, 2n}] // Flatten (* Jean-François Alcover, May 05 2023 *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Yosu Yurramendi, May 24 2013
EXTENSIONS
Definition corrected by María Merino, May 19 2017
STATUS
approved