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

%I A159916 #22 Jul 31 2018 16:00:28
%S A159916 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,
%T A159916 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,
%U A159916 30,80,160,236,236,160,80,30,6,0,6,36,110,240,396,472,396,240,110,36,6,0
%N A159916 Triangle T(m,n) = number of subsets of {1,...,m} with n elements having an odd sum, 1 <= n <= m.
%C A159916 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.
%H A159916 Alois P. Heinz, <a href="/A159916/b159916.txt">Rows n = 1..141, flattened</a>
%H A159916 Johann Cigler, <a href="https://arxiv.org/abs/1711.03340">Some remarks on Rogers-Szegö polynomials and Losanitsch's triangle</a>, arXiv:1711.03340 [math.CO], 2017.
%H A159916 Johann Cigler, <a href="https://homepage.univie.ac.at/johann.cigler/preprints/losanitsch3.pdf">Some Pascal-like triangles</a>, 2018.
%H A159916 Project Euler, <a href="http://projecteuler.net/index.php?section=problems&amp;id=242">Problem 242: Odd Triplets</a>, April 25, 2009.
%F A159916 T(m,m) = A133872(m-1), T(m,1) = A004526(m+1).
%F A159916 T(n,k) = A007318(n,k) - A282011(n,k). - _Alois P. Heinz_, Feb 06 2017
%e A159916 The triangle starts:
%e A159916 (m=1) 1,
%e A159916 (m=2) 1,1,
%e A159916 (m=3) 2,2,0,
%e A159916 (m=4) 2,4,2,0,
%e A159916 (m=5) 3,6,4,2,1,
%e A159916 ...
%e A159916 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}.
%p A159916 b:= proc(n, s) option remember; expand(
%p A159916       `if`(n=0, s, b(n-1, s)+x*b(n-1, irem(s+n, 2))))
%p A159916     end:
%p A159916 T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n, 0)):
%p A159916 seq(T(n), n=1..15);  # _Alois P. Heinz_, Feb 04 2017
%t A159916 b[n_, s_] := b[n, s] = Expand[If[n==0, s, b[n-1, s] + x*b[n-1, Mod[s+n, 2]] ]];
%t A159916 T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 1, n}]][b[n, 0]];
%t A159916 Table[T[n], {n, 1, 15}] // Flatten (* _Jean-François Alcover_, Nov 17 2017, after _Alois P. Heinz_ *)
%o A159916 (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 )
%Y A159916 Cf. A004526, A007318, A133872, A282011.
%Y A159916 T(2n,n) gives A110145.
%K A159916 nonn,tabl
%O A159916 1,4
%A A159916 _M. F. Hasler_, Apr 30 2009