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 A327769 Number of proper twice partitions of n. 4

%I

%S 0,0,0,1,6,15,45,93,223,444,944,1802,3721,6898,13530,25150,48047,

%T 87702,165173,298670,553292,995698,1815981,3242921,5872289,10406853,

%U 18630716,32879716,58391915,102371974,180622850,314943742,551841083,958011541,1667894139

%N Number of proper twice partitions of n.

%H Alois P. Heinz, <a href="/A327769/b327769.txt">Table of n, a(n) for n = 0..5000</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>

%F From _Vaclav Kotesovec_, May 27 2020: (Start)

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

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

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

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

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

%F (End)

%e a(3) = 1:

%e 3 -> 21 -> 111

%e a(4) = 6:

%e 4 -> 31 -> 211

%e 4 -> 31 -> 1111

%e 4 -> 22 -> 112

%e 4 -> 22 -> 211

%e 4 -> 22 -> 1111

%e 4 -> 211-> 1111

%p b:= proc(n, i, k) option remember; `if`(n=0 or k=0, 1, `if`(i>1,

%p b(n, i-1, k), 0) +b(i\$2, k-1)*b(n-i, min(n-i, i), k))

%p end:

%p a:= n-> (k-> add(b(n\$2, i)*(-1)^(k-i)*binomial(k, i), i=0..k))(2):

%p seq(a(n), n=0..37);

%t 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]];

%t a[n_] := Sum[b[n, n, i] (-1)^(2 - i) Binomial[2, i], {i, 0, 2}];

%t a /@ Range[0, 37] (* _Jean-François Alcover_, May 01 2020, after Maple *)

%Y Column k=2 of A327639.

%Y Cf. A063834, A328042.

%K nonn

%O 0,5

%A _Alois P. Heinz_, Sep 24 2019

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Last modified May 8 11:30 EDT 2021. Contains 343666 sequences. (Running on oeis4.)