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A242498
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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.
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14
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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
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OFFSET
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0,7
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COMMENTS
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T(n,k) = T(n+k,-k).
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LINKS
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EXAMPLE
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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;
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MAPLE
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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);
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MATHEMATICA
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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 *)
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CROSSREFS
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Columns k=0-10 gives: A098123, A242499, A242500, A242501, A242502, A242503, A242504, A242505, A242506, A242507, A242508.
Row lengths give A016777(floor(n/2)).
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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