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 A242626 Number T(n,k) of compositions 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. 13
 1, 1, 1, 0, 1, 2, 2, 2, 3, 1, 2, 11, 2, 3, 2, 2, 14, 8, 6, 6, 33, 14, 11, 5, 15, 43, 45, 20, 44, 82, 99, 25, 6, 14, 74, 141, 230, 41, 12, 202, 260, 451, 85, 26, 6, 22, 351, 514, 953, 148, 54, 24, 766, 1049, 1798, 355, 104, 18, 104, 1301, 2321, 3503, 751, 194 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS T(n^2,n) = T(n^2+n,-n) = n! = A000142(n) for n>=0. LINKS Alois P. Heinz, Rows n = 0..500, flattened EXAMPLE T(8,-1) = 15: [2,2,2,2], [1,1,2,4], [1,1,4,2], [1,2,1,4], [1,2,4,1], [1,4,1,2], [1,4,2,1], [2,1,1,4], [2,1,4,1], [2,4,1,1], [4,1,1,2], [4,1,2,1], [4,2,1,1], [4,4], [8]. Triangle T(n,k) begins: : n\k : -3   -2    -1     0     1    2    3 ... +-----+------------------------------------ :  0  :                   1; :  1  :                         1; :  2  :             1,    0,    1; :  3  :                   2,    2; :  4  :             2,    3,    1,   2; :  5  :                  11,    2,   3; :  6  :       2,    2,   14,    8,   6; :  7  :             6,   33,   14,  11; :  8  :       5,   15,   43,   45,  20; :  9  :            44,   82,   99,  25,   6; : 10  :      14,   74,  141,  230,  41,  12; : 11  :           202,  260,  451,  85,  26; : 12  :  6,  22,  351,  514,  953, 148,  54; : 13  :      24,  766, 1049, 1798, 355, 104; : 14  : 18, 104, 1301, 2321, 3503, 751, 194; MAPLE b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0,       expand(add(`if`(j=0, 1, x^(2*irem(i, 2)-1))*       b(n-i*j, i-1, p+j)/j!, j=0..n/i))))     end: T:= n->(p->seq(coeff(p, x, i), i=ldegree(p)..degree(p)))(b(n\$2, 0)): seq(T(n), n=0..20); MATHEMATICA b[n_, i_, p_] := b[n, i, p] = If[n==0, p!, If[i<1, 0, Expand[Sum[If[j==0, 1, x^(2*Mod[i, 2]-1)]*b[n-i*j, i-1, p+j]/j!, {j, 0, n/i}]]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, Exponent[p, x, Min], Exponent[p, x]}]][b[n, n, 0]]; Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, Jan 17 2017, translated from Maple *) CROSSREFS Columns k=(-5)-5 give: A242836, A242837, A242838, A242839, A242840, A242821, A242841, A242842, A242843, A242844, A242845. Row sums give A011782. Cf. A242498 (compositions with multiplicity), A242618 (partitions without multiplicity). Sequence in context: A209254 A227738 A103960 * A240689 A233567 A141059 Adjacent sequences:  A242623 A242624 A242625 * A242627 A242628 A242629 KEYWORD nonn,tabf AUTHOR Alois P. Heinz, May 19 2014 STATUS approved

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Last modified August 19 08:17 EDT 2019. Contains 326115 sequences. (Running on oeis4.)