A328041 Number of parts in all proper floor(n/2)-times partitions of n.

0, 1, 2, 5, 21, 61, 461, 1652, 17487, 76264, 1002835, 5207742, 88664398, 515821495, 10184805624, 69200406679, 1610282904928, 12024183111167, 318978837371853, 2653055962437988, 79332250069994262, 725413309833320933, 23919660963588169669, 238830233430136549070

Offset: 0

Author

Alois P. Heinz, Oct 02 2019

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A327631.

Programs

Maple

b:= proc(n, i, k) option remember; `if`(n=0, [1, 0],
     `if`(k=0, [1, 1], `if`(i<2, 0, b(n, i-1, k))+
         (h-> (f-> f +[0, f[1]*h[2]/h[1]])(h[1]*
        b(n-i, min(n-i, i), k)))(b(i$2, k-1))))
    end:
a:= n-> (k-> add(b(n$2, i)[2]*(-1)^(k-i)
        *binomial(k, i), i=0..k))(iquo(n,2)):
seq(a(n), n=0..23);

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[n==0, {1, 0}, If[k==0, {1, 1}, If[i<2, 0, b[n, i - 1, k]] + Function[h, Function[f, f + {0, f[[1]] h[[2]]/h[[1]]}][h[[1]] b[n - i, Min[n - i, i], k]]][b[i, i, k - 1]]]];
a[n_] := With[{k = Quotient[n, 2]}, Sum[b[n, n, i][[2]] (-1)^(k - i)* Binomial[k, i], {i, 0, k}]];
a /@ Range[0, 23] (* Jean-François Alcover, Dec 18 2020, after Alois P. Heinz *)

Formula

a(n) = A327631(n,floor(n/2)).