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A327769 Number of proper twice partitions of n. 4
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified April 17 13:31 EDT 2021. Contains 343063 sequences. (Running on oeis4.)