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

1,4

Links

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

Example

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

Maple

b:= proc(n, i, k) option remember; `if`(n=0, [1, 0],
     `if`(k=0, [1, 1], `if`(i*(i+1)/2<n, 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-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 *)

Crossrefs

Column k=2 of A327632.

Cf. A327605.

Sequence in context: A005313 A213674 A108639 * A087218 A098075 A137584

Adjacent sequences: A327792 A327793 A327794 * A327796 A327797 A327798

Keyword

nonn

Author

Alois P. Heinz, Sep 25 2019

Status

approved