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

0,5

Links

Alois P. Heinz, Table of n, a(n) for n = 0..5000

Wikipedia, Partition (number theory)

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)

Example

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

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 *)

Crossrefs

Column k=2 of A327639.

Cf. A063834, A328042.

Sequence in context: A197160 A182420 A117961 * A318482 A095122 A215917

Adjacent sequences: A327766 A327767 A327768 * A327770 A327771 A327772

Keyword

nonn

Author

Alois P. Heinz, Sep 24 2019

Status

approved