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A327769 Number of proper twice partitions of n.

%I A327769 #18 May 27 2020 10:54:52
%S A327769 0,0,0,1,6,15,45,93,223,444,944,1802,3721,6898,13530,25150,48047,
%T A327769 87702,165173,298670,553292,995698,1815981,3242921,5872289,10406853,
%U A327769 18630716,32879716,58391915,102371974,180622850,314943742,551841083,958011541,1667894139
%N A327769 Number of proper twice partitions of n.
%H A327769 Alois P. Heinz, <a href="/A327769/b327769.txt">Table of n, a(n) for n = 0..5000</a>
%H A327769 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%F A327769 From _Vaclav Kotesovec_, May 27 2020: (Start)
%F A327769 a(n) ~ c * 5^(n/4), where
%F A327769 c = 96146522937.7161... if mod(n,4) = 0
%F A327769 c = 96146521894.9433... if mod(n,4) = 1
%F A327769 c = 96146522937.2138... if mod(n,4) = 2
%F A327769 c = 96146521894.8218... if mod(n,4) = 3
%F A327769 (End)
%e A327769 a(3) = 1:
%e A327769   3 -> 21 -> 111
%e A327769 a(4) = 6:
%e A327769   4 -> 31 -> 211
%e A327769   4 -> 31 -> 1111
%e A327769   4 -> 22 -> 112
%e A327769   4 -> 22 -> 211
%e A327769   4 -> 22 -> 1111
%e A327769   4 -> 211-> 1111
%p A327769 b:= proc(n, i, k) option remember; `if`(n=0 or k=0, 1, `if`(i>1,
%p A327769       b(n, i-1, k), 0) +b(i$2, k-1)*b(n-i, min(n-i, i), k))
%p A327769     end:
%p A327769 a:= n-> (k-> add(b(n$2, i)*(-1)^(k-i)*binomial(k, i), i=0..k))(2):
%p A327769 seq(a(n), n=0..37);
%t A327769 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 A327769 a[n_] := Sum[b[n, n, i] (-1)^(2 - i) Binomial[2, i], {i, 0, 2}];
%t A327769 a /@ Range[0, 37] (* _Jean-François Alcover_, May 01 2020, after Maple *)
%Y A327769 Column k=2 of A327639.
%Y A327769 Cf. A063834, A328042.
%K A327769 nonn
%O A327769 0,5
%A A327769 _Alois P. Heinz_, Sep 24 2019