OFFSET
1,4
COMMENTS
One could extend the triangle to include values for m=0 and/or n=0, but these correspond to empty sets and would always be 0. The first odd value for odd m and 1<n<m is T(13,5) = 651.
LINKS
Alois P. Heinz, Rows n = 1..141, flattened
Johann Cigler, Some remarks on Rogers-Szegö polynomials and Losanitsch's triangle, arXiv:1711.03340 [math.CO], 2017.
Johann Cigler, Some Pascal-like triangles, 2018.
Project Euler, Problem 242: Odd Triplets, April 25, 2009.
FORMULA
EXAMPLE
The triangle starts:
(m=1) 1,
(m=2) 1,1,
(m=3) 2,2,0,
(m=4) 2,4,2,0,
(m=5) 3,6,4,2,1,
...
T(5,3)=4, since the set {1,2,3,4,5} has four 3-element subsets having an odd sum of elements, namely {1,2,4}, {1,3,5}, {2,3,4} and {2,4,5}.
MAPLE
b:= proc(n, s) option remember; expand(
`if`(n=0, s, b(n-1, s)+x*b(n-1, irem(s+n, 2))))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n, 0)):
seq(T(n), n=1..15); # Alois P. Heinz, Feb 04 2017
MATHEMATICA
b[n_, s_] := b[n, s] = Expand[If[n==0, s, b[n-1, s] + x*b[n-1, Mod[s+n, 2]] ]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 1, n}]][b[n, 0]];
Table[T[n], {n, 1, 15}] // Flatten (* Jean-François Alcover, Nov 17 2017, after Alois P. Heinz *)
PROG
(PARI) T(n, k)=sum( i=2^k-1, 2^n-2^(n-k), norml2(binary(i))==k & sum(j=0, n\2, bittest(i, 2*j))%2 )
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
M. F. Hasler, Apr 30 2009
STATUS
approved