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 A242498 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; triangle T(n,k), n>=0, -floor(n/2)+(n mod 2)<=k<=n, read by rows. 14
 1, 1, 1, 0, 0, 1, 2, 1, 0, 1, 1, 1, 0, 3, 2, 0, 1, 3, 4, 1, 4, 3, 0, 1, 1, 2, 1, 6, 9, 3, 5, 4, 0, 1, 4, 9, 6, 11, 16, 6, 6, 5, 0, 1, 1, 3, 3, 11, 24, 18, 19, 25, 10, 7, 6, 0, 1, 5, 16, 18, 28, 51, 40, 31, 36, 15, 8, 7, 0, 1, 1, 4, 6, 19, 51, 60, 65, 95, 75, 48, 49, 21, 9, 8, 0, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,7 COMMENTS T(n,k) = T(n+k,-k). LINKS Alois P. Heinz, Rows n = 0..120, flattened EXAMPLE Triangle T(n,k) begins: : n\k : -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 ... +-----+--------------------------------------------------------- : 0 : 1; : 1 : 1; : 2 : 1, 0, 0, 1; : 3 : 2, 1, 0, 1; : 4 : 1, 1, 0, 3, 2, 0, 1; : 5 : 3, 4, 1, 4, 3, 0, 1; : 6 : 1, 2, 1, 6, 9, 3, 5, 4, 0, 1; : 7 : 4, 9, 6, 11, 16, 6, 6, 5, 0, 1; : 8 : 1, 3, 3, 11, 24, 18, 19, 25, 10, 7, 6, 0, 1; : 9 : 5, 16, 18, 28, 51, 40, 31, 36, 15, 8, 7, 0, 1; : 10 : 1, 4, 6, 19, 51, 60, 65, 95, 75, 48, 49, 21, 9, 8, 0, 1; MAPLE b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0, expand( add(x^(j*(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[x^(j*(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, Feb 11 2015, after Alois P. Heinz *) CROSSREFS Columns k=0-10 gives: A098123, A242499, A242500, A242501, A242502, A242503, A242504, A242505, A242506, A242507, A242508. Row sums give A011782. Diagonals include: A000012, A000004, A001477, A000217, A000290, A180415. Row lengths give A016777(floor(n/2)). Cf. A240009, A240021. Sequence in context: A088705 A065712 A153172 * A321927 A016194 A258139 Adjacent sequences: A242495 A242496 A242497 * A242499 A242500 A242501 KEYWORD nonn,tabf AUTHOR Alois P. Heinz, May 16 2014 STATUS approved

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Last modified September 12 04:22 EDT 2024. Contains 375842 sequences. (Running on oeis4.)