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A320545 Number of partitions of n into parts of exactly three sorts which are introduced in ascending order such that sorts of adjacent parts are different.

%I A320545 #11 Dec 08 2020 08:05:10
%S A320545 1,4,12,31,73,165,357,760,1582,3270,6678,13589,27482,55468,111588,
%T A320545 224259,449908,902106,1807173,3619162,7244557,14499238,29011551,
%U A320545 58044194,116115782,232275383,464607730,929306306,1858730674,3717648658,7435541392,14871467784
%N A320545 Number of partitions of n into parts of exactly three sorts which are introduced in ascending order such that sorts of adjacent parts are different.
%H A320545 Alois P. Heinz, <a href="/A320545/b320545.txt">Table of n, a(n) for n = 3..1000</a>
%F A320545 a(n) ~ 2^(n-2) / QPochhammer[1/2]. - _Vaclav Kotesovec_, Oct 25 2018
%p A320545 b:= proc(n, i, k) option remember; `if`(n=0 or i=1, k^(n-1),
%p A320545       b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k)))
%p A320545     end:
%p A320545 A:= (n, k)-> `if`(n=0, 1, `if`(k<2, k, k*b(n$2, k-1))):
%p A320545 a:= n-> (k-> add(A(n, k-i)*(-1)^i/(i!*(k-i)!), i=0..k))(3):
%p A320545 seq(a(n), n=3..40);
%t A320545 b[n_, i_, k_] := b[n, i, k] = If[n == 0 || i == 1, k^(n - 1), b[n, i - 1, k] + If[i > n, 0, k b[n - i, i, k]]];
%t A320545 A[n_, k_] := If[n == 0, 1, If[k < 2, k, k b[n, n, k - 1]]];
%t A320545 a[n_] := With[{k = 3}, Sum[A[n, k - i] (-1)^i/(i! (k - i)!), {i, 0, k}]];
%t A320545 a /@ Range[3, 40] (* _Jean-François Alcover_, Dec 08 2020, after _Alois P. Heinz_ *)
%Y A320545 Column k=3 of A262495.
%Y A320545 Cf. A258458, A262445.
%K A320545 nonn
%O A320545 3,2
%A A320545 _Alois P. Heinz_, Oct 15 2018