cp's OEIS Frontend

A321633 Number of permutations of the multiset {1,1,1,1,2,2,2,2,3,3,3,3,...,n,n,n,n} with no two consecutive terms equal.

%I A321633 #27 Sep 29 2023 13:05:50
%S A321633 1,0,2,1092,2265024,11804626080,131402141197200,2778291737177034960,
%T A321633 102284730928300590754560,6134232798447803932455457920,
%U A321633 568598490353320413296928514444800,78076149156802562231395694989534464000,15336188146163145199585928509793662920345600
%N A321633 Number of permutations of the multiset {1,1,1,1,2,2,2,2,3,3,3,3,...,n,n,n,n} with no two consecutive terms equal.
%H A321633 Seiichi Manyama, <a href="/A321633/b321633.txt">Table of n, a(n) for n = 0..129</a>
%H A321633 Mathematics.StackExchange, <a href="https://math.stackexchange.com/questions/129451/find-the-number-of-arrangements-of-k-mbox-1s-k-mbox-2s-cdots">Find the number of k 1's, k 2's, ... , k n's - total kn cards</a>, Apr 08 2012.
%F A321633 a(n) = n! * A190830(n).
%F A321633 a(n) = Integral_{0..oo} (-x + 3/2 * x^2 - 1/2 * x^3 + 1/24 * x^4)^n * exp(-x) dx.
%e A321633 a(2) = 2 because there are two permutations of {1,1,1,1,2,2,2,2} avoiding equal consecutive terms: 12121212 and 21212121.
%t A321633 a[n_] := Integrate[(-x + 3/2 * x^2 - 1/2 * x^3 + 1/24 * x^4)^n  * Exp[-x],  {x, 0, Infinity}]; Array[a, 10, 0] (* _Stefano Spezia_, Nov 27 2018 *)
%Y A321633 Row 4 of A322093.
%Y A321633 Cf. A000142, A114938, A190830, A193638.
%K A321633 nonn
%O A321633 0,3
%A A321633 _Seiichi Manyama_, Nov 15 2018