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%I #16 Oct 27 2023 20:47:59
%S 0,1,0,2,0,3,1,5,1,8,3,12,4,18,7,27,10,38,16,54,22,76,33,104,45,142,
%T 64,192,87,256,120,340,159,448,215,585,283,760,374,982,486,1260,634,
%U 1610,814,2048,1049,2590,1335,3264,1700,4097,2146,5120,2708,6378,3390,7917,4243,9792,5276
%N The difference between the number of partitions of n into odd parts (A000009) and the number of partitions of n into even parts (A035363).
%H Alois P. Heinz, <a href="/A282892/b282892.txt">Table of n, a(n) for n = 0..10000</a> (terms n=1..165 from Robert G. Wilson v)
%F a(2n-1) = A000009(2n-1) = A078408(n).
%F a(2n) = A282893(n).
%p with(numtheory):
%p b:= proc(n, t) option remember; `if`(n=0, 1, add(add(`if`(
%p (d+t)::odd, d, 0), d=divisors(j))*b(n-j, t), j=1..n)/n)
%p end:
%p a:= n-> b(n, 0) -b(n, 1):
%p seq(a(n), n=0..80); # _Alois P. Heinz_, Feb 24 2017
%t f[n_] := Length[ IntegerPartitions[n, All, 2Range[n] -1]] - Length[ IntegerPartitions[n, All, 2 Range[n]]]; Array[f, 60]
%t (* Second program: *)
%t b[n_, t_] := b[n, t] = If[n == 0, 1, Sum[Sum[If[
%t OddQ[d+t], d, 0], {d, Divisors[j]}]*b[n-j, t], {j, 1, n}]/n];
%t a[n_] := b[n, 0] - b[n, 1];
%t a /@ Range[0, 80] (* _Jean-François Alcover_, Jun 06 2021, after _Alois P. Heinz_ *)
%Y Cf. A000009, A035363, A078408, A282893.
%K nonn
%O 0,4
%A _Robert G. Wilson v_, Feb 24 2017
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