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Number of compositions of n in which each part has odd multiplicity.
3

%I #13 Feb 08 2017 12:48:39

%S 1,1,1,4,3,10,16,28,49,91,186,266,670,884,2350,3028,8259,10536,30241,

%T 37382,108628,135550,391202,503750,1429838,1884659,5222976,7107138,

%U 19119324,27088726,70366026,103884570,259884905,399686188,962312254,1543116240,3576132805

%N Number of compositions of n in which each part has odd multiplicity.

%H Alois P. Heinz, <a href="/A242391/b242391.txt">Table of n, a(n) for n = 0..1000</a>

%e a(0) = 1: the empty composition.

%e a(1) = 1: [1].

%e a(2) = 1: [2].

%e a(3) = 4: [3], [2,1], [1,2], [1,1,1].

%e a(4) = 3: [4], [3,1], [1,3].

%e a(5) = 10: [5], [4,1], [1,4], [3,2], [2,3], [2,1,1,1], [1,2,1,1], [1,1,2,1], [1,1,1,2], [1,1,1,1,1].

%p b:= proc(n, i, p) option remember; `if`(n=0, p!,

%p `if`(i<1, 0, add(`if`(j=0 or irem(j, 2)=1,

%p b(n-i*j, i-1, p+j)/j!, 0), j=0..n/i)))

%p end:

%p a:= n-> b(n$2, 0):

%p seq(a(n), n=0..45);

%t b[n_, i_, p_] := b[n, i, p] = If[n==0, p!, If[i<1, 0, Sum[If[j==0 || Mod[j, 2]==1, b[n-i*j, i-1, p+j]/j!, 0], {j, 0, n/i}]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 45}] (* _Jean-François Alcover_, Feb 08 2017, translated from Maple *)

%Y Cf. A130495 (for even multiplicity).

%K nonn

%O 0,4

%A _Alois P. Heinz_, May 12 2014