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A320547 Number of partitions of n into parts of exactly five sorts which are introduced in ascending order such that sorts of adjacent parts are different.

%I A320547 #9 Dec 08 2020 08:36:53
%S A320547 1,11,77,438,2216,10423,46732,202826,860599,3593651,14835058,60735635,
%T A320547 247155920,1001321100,4043485479,16288776186,65500040622,263035896496,
%U A320547 1055252507399,4230340498375,16949360224358,67881450386237,271777857121332,1087867654290457
%N A320547 Number of partitions of n into parts of exactly five sorts which are introduced in ascending order such that sorts of adjacent parts are different.
%H A320547 Alois P. Heinz, <a href="/A320547/b320547.txt">Table of n, a(n) for n = 5..1000</a>
%F A320547 a(n) ~ 4^(n-1) / (4! * QPochhammer[1/4]). - _Vaclav Kotesovec_, Oct 25 2018
%p A320547 b:= proc(n, i, k) option remember; `if`(n=0 or i=1, k^(n-1),
%p A320547       b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k)))
%p A320547     end:
%p A320547 A:= (n, k)-> `if`(n=0, 1, `if`(k<2, k, k*b(n$2, k-1))):
%p A320547 a:= n-> (k-> add(A(n, k-i)*(-1)^i/(i!*(k-i)!), i=0..k))(5):
%p A320547 seq(a(n), n=5..40);
%t A320547 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 A320547 A[n_, k_] := If[n == 0, 1, If[k < 2, k, k b[n, n, k - 1]]];
%t A320547 a[n_] := With[{k = 5}, Sum[A[n, k - i] (-1)^i/(i! (k - i)!), {i, 0, k}]];
%t A320547 a /@ Range[5, 40] (* _Jean-François Alcover_, Dec 08 2020, after _Alois P. Heinz_ *)
%Y A320547 Column k=5 of A262495.
%Y A320547 Cf. A258460.
%K A320547 nonn
%O A320547 5,2
%A A320547 _Alois P. Heinz_, Oct 15 2018