A282011 Number T(n,k) of k-element subsets of [n] having an even sum; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 2, 4, 6, 3, 0, 1, 3, 6, 10, 9, 3, 0, 1, 3, 9, 19, 19, 9, 3, 1, 1, 4, 12, 28, 38, 28, 12, 4, 1, 1, 4, 16, 44, 66, 60, 40, 20, 5, 0, 1, 5, 20, 60, 110, 126, 100, 60, 25, 5, 0, 1, 5, 25, 85, 170, 226, 226, 170, 85, 25, 5, 1, 1, 6, 30, 110, 255, 396, 452, 396, 255, 110, 30, 6, 1
Offset: 0
Examples
T(5,0) = 1: {}.
T(5,1) = 2: {2}, {4}.
T(5,2) = 4: {1,3}, {1,5}, {2,4}, {3,5}.
T(5,3) = 6: {1,2,3}, {1,2,5}, {1,3,4}, {1,4,5}, {2,3,5}, {3,4,5}.
T(5,4) = 3: {1,2,3,4}, {1,2,4,5}, {2,3,4,5}.
T(5,5) = 0.
T(7,7) = 1: {1,2,3,4,5,6,7}.
Triangle T(n,k) begins:
1;
1, 0;
1, 1, 0;
1, 1, 1, 1;
1, 2, 2, 2, 1;
1, 2, 4, 6, 3, 0;
1, 3, 6, 10, 9, 3, 0;
1, 3, 9, 19, 19, 9, 3, 1;
1, 4, 12, 28, 38, 28, 12, 4, 1;
1, 4, 16, 44, 66, 60, 40, 20, 5, 0;
1, 5, 20, 60, 110, 126, 100, 60, 25, 5, 0;
1, 5, 25, 85, 170, 226, 226, 170, 85, 25, 5, 1;
1, 6, 30, 110, 255, 396, 452, 396, 255, 110, 30, 6, 1;
Links
- Alois P. Heinz, Rows n = 0..200, 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.
Crossrefs
Columns k=0..10 give (offsets may differ): A000012, A004526, A002620, A005993, A005994, A032092, A032093, A018211, A018212, A282077, A282078.
Row sums give A011782.
Main diagonal gives A133872(n+1).
Lower diagonals T(n+j,n) for j=1..10 give: A004525(n+1), A282079, A228705, A282080, A282081, A282082, A282083, A282084, A282085, A282086.
T(2n,n) gives A119358.
Programs
-
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=0..n))(b(n, 1)): seq(T(n), n=0..16); -
Mathematica
Flatten[Table[Sum[Binomial[Ceiling[n/2],2j]Binomial[Floor[n/2],k-2j],{j,0,Floor[(n+1)/4]}],{n,0,10},{k,0,n}]] (* Indranil Ghosh, Feb 26 2017 *) -
PARI
a(n,k)=sum(j=0,floor((n+1)/4),binomial(ceil(n/2),2*j)*binomial(floor(n/2),k-2*j)); tabl(nn)={for(n=0,nn,for(k=0,n,print1(a(n,k),", "););print(););} \\ Indranil Ghosh, Feb 26 2017
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