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 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. 20
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,12 COMMENTS Row n is symmetric if and only if n mod 4 in {0,3} (or if T(n,n) = 1). LINKS Alois P. Heinz, Rows n = 0..200, flattened Johann Cigler, Some remarks on Rogers-SzegĂ¶ polynomials and Losanitsch's triangle, arXiv:1711.03340 [math.CO], 2017. Johann Cigler, Some Pascal-like triangles, 2018. FORMULA T(n,k) = Sum_{j=0..floor((n+1)/4)} C(ceiling(n/2),2*j) * C(floor(n/2),k-2*j). T(n,k) = A007318(n,k) - A159916(n,k). 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]. EXAMPLE 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; 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 *) PROG (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 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 diaginal 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. Cf. A007318, A057711, A087447, A159916. Sequence in context: A078826 A051950 A172353 * A245514 A104754 A206827 Adjacent sequences:  A282008 A282009 A282010 * A282012 A282013 A282014 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Feb 04 2017 STATUS approved

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Last modified January 17 05:26 EST 2019. Contains 319207 sequences. (Running on oeis4.)