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A320546 Number of partitions of n into parts of exactly four sorts which are introduced in ascending order such that sorts of adjacent parts are different.

%I A320546 #9 Dec 08 2020 08:04:51
%S A320546 1,7,33,130,464,1558,5039,15886,49282,151165,460352,1394863,4212752,
%T A320546 12694566,38197710,114820403,344919283,1035670246,3108844526,
%U A320546 9330186438,27997888759,84008273161,252054096569,756220672185,2268778953179,6806570182252,20420177671614
%N A320546 Number of partitions of n into parts of exactly four sorts which are introduced in ascending order such that sorts of adjacent parts are different.
%H A320546 Alois P. Heinz, <a href="/A320546/b320546.txt">Table of n, a(n) for n = 4..1000</a>
%F A320546 a(n) ~ 3^(n-1) / (3! * QPochhammer[1/3]). - _Vaclav Kotesovec_, Oct 25 2018
%p A320546 b:= proc(n, i, k) option remember; `if`(n=0 or i=1, k^(n-1),
%p A320546       b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k)))
%p A320546     end:
%p A320546 A:= (n, k)-> `if`(n=0, 1, `if`(k<2, k, k*b(n$2, k-1))):
%p A320546 a:= n-> (k-> add(A(n, k-i)*(-1)^i/(i!*(k-i)!), i=0..k))(4):
%p A320546 seq(a(n), n=4..40);
%t A320546 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 A320546 A[n_, k_] := If[n == 0, 1, If[k < 2, k, k b[n, n, k - 1]]];
%t A320546 a[n_] := With[{k = 4}, Sum[A[n, k - i] (-1)^i/(i! (k - i)!), {i, 0, k}]];
%t A320546 a /@ Range[4, 40] (* _Jean-François Alcover_, Dec 08 2020, after _Alois P. Heinz_ *)
%Y A320546 Column k=4 of A262495.
%Y A320546 Cf. A258459.
%K A320546 nonn
%O A320546 4,2
%A A320546 _Alois P. Heinz_, Oct 15 2018