A327769 Number of proper twice partitions of n.

0, 0, 0, 1, 6, 15, 45, 93, 223, 444, 944, 1802, 3721, 6898, 13530, 25150, 48047, 87702, 165173, 298670, 553292, 995698, 1815981, 3242921, 5872289, 10406853, 18630716, 32879716, 58391915, 102371974, 180622850, 314943742, 551841083, 958011541, 1667894139

Offset: 0

Author

Alois P. Heinz, Sep 24 2019

Keywords

• nonn

Examples

			a(3) = 1:
  3 -> 21 -> 111
a(4) = 6:
  4 -> 31 -> 211
  4 -> 31 -> 1111
  4 -> 22 -> 112
  4 -> 22 -> 211
  4 -> 22 -> 1111
  4 -> 211-> 1111
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..5000
• Wikipedia, Partition (number theory)

Crossrefs

Column k=2 of A327639.

Cf. A063834, A328042.

Programs

Maple

b:= proc(n, i, k) option remember; `if`(n=0 or k=0, 1, `if`(i>1,
      b(n, i-1, k), 0) +b(i$2, k-1)*b(n-i, min(n-i, i), k))
    end:
a:= n-> (k-> add(b(n$2, i)*(-1)^(k-i)*binomial(k, i), i=0..k))(2):
seq(a(n), n=0..37);

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[n == 0 || k == 0, 1, If[i > 1, b[n, i - 1, k], 0] + b[i, i, k - 1] b[n - i, Min[n - i, i], k]];
a[n_] := Sum[b[n, n, i] (-1)^(2 - i) Binomial[2, i], {i, 0, 2}];
a /@ Range[0, 37] (* Jean-François Alcover, May 01 2020, after Maple *)

Formula

From Vaclav Kotesovec, May 27 2020: (Start)

a(n) ~ c * 5^(n/4), where

c = 96146522937.7161... if mod(n,4) = 0

c = 96146521894.9433... if mod(n,4) = 1

c = 96146522937.2138... if mod(n,4) = 2

c = 96146521894.8218... if mod(n,4) = 3

(End)