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A282011
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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.
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20
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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
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OFFSET
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0,12
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COMMENTS
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Row n is symmetric if and only if n mod 4 in {0,3} (or if T(n,n) = 1).
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LINKS
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FORMULA
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T(n,k) = Sum_{j=0..floor((n+1)/4)} C(ceiling(n/2),2*j) * C(floor(n/2),k-2*j).
Sum_{k=0..n} k * T(n,k) = A057711(n-1) for n>0.
Sum_{k=0..n} (k+1) * T(n,k) = A087447(n) + [n=2].
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EXAMPLE
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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;
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MAPLE
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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);
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MATHEMATICA
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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 *)
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PROG
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(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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CROSSREFS
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Columns k=0..10 give (offsets may differ): A000012, A004526, A002620, A005993, A005994, A032092, A032093, A018211, A018212, A282077, A282078.
Lower diagonals T(n+j,n) for j=1..10 give: A004525(n+1), A282079, A228705, A282080, A282081, A282082, A282083, A282084, A282085, A282086.
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KEYWORD
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AUTHOR
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STATUS
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approved
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