A327795 Number of parts in all proper twice partitions of n into distinct parts.

0, 0, 0, 3, 6, 13, 30, 61, 121, 210, 353, 600, 989, 1628, 2667, 4205, 6514, 10406, 15893, 24322, 37516, 56824, 85102, 128420, 191579, 284898, 422839, 622721, 913006, 1345320, 1958269, 2843788, 4140170, 5983662, 8632808, 12433730, 17830728, 25527909, 36516161

Offset: 1

Author

Alois P. Heinz, Sep 25 2019

Keywords

• nonn

Examples

			a(4) = 3:
  4 -> 31 -> 211   (3 parts)
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..5000

Crossrefs

Column k=2 of A327632.

Cf. A327605.

Programs

Maple

b:= proc(n, i, k) option remember; `if`(n=0, [1, 0],
     `if`(k=0, [1, 1], `if`(i*(i+1)/2 (f-> f +[0, f[1]*h[2]/h[1]])(h[1]*
        b(n-i, min(n-i, i-1), 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))(2):
seq(a(n), n=1..41);

Mathematica

b[n_, i_, k_] := b[n, i, k] = With[{}, If[n == 0, {1, 0}, If[k == 0, {1, 1}, If[i (i + 1)/2 < n, {0, 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 - 1], k]]][ b[i, i, k - 1]]]]]];
T[n_, k_] := Sum[b[n, n, i][[2]] (-1)^(k - i) Binomial[k, i], {i, 0, k}];
a[n_] := T[n, 2];
Array[a, 41] (* Jean-François Alcover, Dec 09 2020, after Alois P. Heinz *)