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 A242618 Number T(n,k) of partitions of n, where k is the difference between the number of odd parts and the number of even parts, both counted without multiplicity; triangle T(n,k), n>=0, read by rows. 28
 1, 1, 1, 0, 1, 1, 2, 2, 1, 1, 1, 4, 2, 1, 1, 2, 3, 3, 2, 1, 8, 3, 3, 2, 4, 6, 5, 5, 4, 13, 8, 4, 1, 5, 5, 11, 13, 7, 1, 11, 20, 14, 9, 2, 1, 6, 13, 17, 26, 11, 3, 1, 22, 31, 27, 15, 5, 2, 12, 18, 34, 44, 18, 7, 4, 40, 47, 51, 23, 11, 5, 16, 36, 56, 72, 34, 11, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,7 COMMENTS T(n,0) = A241638(n). Sum_{k<0} T(n,k) = A241640(n). Sum_{k<=0} T(n,k) = A241639(n). Sum_{k>=0} T(n,k) = A241637(n). Sum_{k>0} T(n,k) = A241636(n). T(n^2,n) = T(n^2+n,-n) = 1. T(n^2+n,n) = Sum_{k} T(n,k) = A000041(n). T(n^2+3*n,-n) = A000712(n). LINKS Alois P. Heinz, Rows n = 0..500, flattened EXAMPLE Triangle T(n,k) begins: : n\k : -3  -2  -1   0   1   2   3 ... +-----+--------------------------- :  0  :              1; :  1  :                  1; :  2  :          1,  0,  1; :  3  :              1,  2; :  4  :          2,  1,  1,  1; :  5  :              4,  2,  1; :  6  :      1,  2,  3,  3,  2; :  7  :          1,  8,  3,  3; :  8  :      2,  4,  6,  5,  5; :  9  :          4, 13,  8,  4,  1; : 10  :      5,  5, 11, 13,  7,  1; : 11  :         11, 20, 14,  9,  2; : 12  :  1,  6, 13, 17, 26, 11,  3; : 13  :      1, 22, 31, 27, 15,  5; : 14  :  2, 12, 18, 34, 44, 18,  7; MAPLE b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,       expand(b(n, i-1)+add(b(n-i*j, i-1)*x^(2*irem(i, 2)-1), j=1..n/i))))     end: T:= n-> (p-> seq(coeff(p, x, i), i=ldegree(p)..degree(p)))(b(n\$2)): seq(T(n), n=0..20); MATHEMATICA b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Expand[b[n, i - 1] + Sum[b[n - i*j, i - 1]*x^(2*Mod[i, 2] - 1), {j, 1, n/i}]]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, Exponent[p, x, Min], Exponent[p, x]}]][b[n, n]]; Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, Dec 12 2016 after Alois P. Heinz *) CROSSREFS Columns k=(-10)-10 give: A242682, A242683, A242684, A242685, A242686, A242687, A242688, A242689, A242690, A242691, A241638, A242692, A242693, A242694, A242695, A242696, A242697, A242698, A242699, A242700, A242701. Row sums give A000041. Cf. A240009 (parts counted with multiplicity), A240021 (distinct parts), A242626 (compositions counted without multiplicity). Sequence in context: A259922 A162741 A104320 * A180264 A225200 A128706 Adjacent sequences:  A242615 A242616 A242617 * A242619 A242620 A242621 KEYWORD nonn,tabf,look AUTHOR Alois P. Heinz, May 19 2014 STATUS approved

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Last modified June 25 12:55 EDT 2019. Contains 324352 sequences. (Running on oeis4.)