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

0,3

Links

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

Formula

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

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 *)

Crossrefs

Cf. A327631.

Sequence in context: A359725 A000941 A000131 * A242785 A359672 A228385

Adjacent sequences: A328038 A328039 A328040 * A328042 A328043 A328044

Keyword

nonn

Author

Alois P. Heinz, Oct 02 2019

Status

approved