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A226048
Irregular triangle read by rows: T(n,k) is the number of binary pattern classes in the (2,n)-rectangular grid with k '1's and (2n-k) '0's: two patterns are in same class if one of them can be obtained by a reflection or 180-degree rotation of the other.
41
1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 6, 6, 6, 2, 1, 1, 2, 10, 14, 22, 14, 10, 2, 1, 1, 3, 15, 32, 60, 66, 60, 32, 15, 3, 1, 1, 3, 21, 55, 135, 198, 246, 198, 135, 55, 21, 3, 1, 1, 4, 28, 94, 266, 508, 777, 868, 777, 508, 266, 94, 28, 4, 1, 1, 4, 36
OFFSET
0,7
COMMENTS
Sum of rows (see example) gives A225826.
This triangle is to A225826 as Losanitsch's triangle A034851 is to A005418.
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