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A331606
Number of compositions of n with the multiplicity of the first part odd.
3
1, 1, 4, 4, 12, 18, 44, 72, 158, 288, 604, 1146, 2332, 4528, 9126, 17944, 35940, 71130, 142132, 282344, 563630, 1121936, 2239060, 4462530, 8906236, 17764160, 35458774, 70761520, 141272876, 282025466, 563159588, 1124543256, 2245918406, 4485670168, 8960061076
OFFSET
1,3
LINKS
M. Archibald, A. Blecher, A. Knopfmacher, M. E. Mays, Inversions and Parity in Compositions of Integers, J. Int. Seq., Vol. 23 (2020), Article 20.4.1.
FORMULA
G.f.: Sum_{i>=1} (1-x)*x^i/(2*(-2*x^(i+1)+2*x^i-2*x+1)) + x/(2*(1-2*x)).
a(n) = A011782(n) - A331609(n). - Alois P. Heinz, Jan 23 2020
EXAMPLE
For n=3, a(4)=4 as we count 4, 3+1, 1+3 and 2+1+1.
MAPLE
b:= proc(n, p, t) option remember; `if`(n=0, t,
add(b(n-j, p, `if`(p=j, 1-t, t)), j=1..n))
end:
a:= n-> add(b(n-j, j, 1), j=1..n):
seq(a(n), n=1..38); # Alois P. Heinz, Jan 23 2020
MATHEMATICA
gf[x_] := x/(2 (1 - 2 x)) + Sum[(1 - x) x^i/(2 (-2 x^(i + 1) + 2 x^i - 2 x + 1)) , {i, 1, 40}]; CL := CoefficientList[Series[gf[x], {x, 0, 35}], x];
Drop[CL, 1] (* Peter Luschny, Jan 23 2020 *)
CROSSREFS
Cf. A011782, A331609 (similar, with even).
Sequence in context: A157617 A053415 A303315 * A079902 A309128 A120033
KEYWORD
nonn
AUTHOR
Arnold Knopfmacher, Jan 22 2020
STATUS
approved