A159916 Triangle T(m,n) = number of subsets of {1,...,m} with n elements having an odd sum, 1 <= n <= m.

1, 1, 1, 2, 2, 0, 2, 4, 2, 0, 3, 6, 4, 2, 1, 3, 9, 10, 6, 3, 1, 4, 12, 16, 16, 12, 4, 0, 4, 16, 28, 32, 28, 16, 4, 0, 5, 20, 40, 60, 66, 44, 16, 4, 1, 5, 25, 60, 100, 126, 110, 60, 20, 5, 1, 6, 30, 80, 160, 236, 236, 160, 80, 30, 6, 0, 6, 36, 110, 240, 396, 472, 396, 240, 110, 36, 6, 0

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, Apr 25 2009.

Formula

T(m,m) = A133872(m-1), T(m,1) = A004526(m+1).

T(n,k) = A007318(n,k) - A282011(n,k). - Alois P. Heinz, Feb 06 2017

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

Cf. A004526, A007318, A133872, A282011.

T(2n,n) gives A110145.

Sequence in context: A171933 A074823 A226177 * A159286 A355837 A389946

Adjacent sequences: A159913 A159914 A159915 * A159917 A159918 A159919

Keyword

nonn,tabl

Author

M. F. Hasler, Apr 30 2009

Status

approved