A293809 - OEIS

%I #13 Nov 23 2023 10:37:52

%S 1,2,7,22,73,240,818,2824,10004,36252,134594,512632,2002797,8037634,

%T 33122211,140287074,610344666,2728599114,12524559427,59014996342,

%U 285169596358,1412357461074,7161541766341,37150562120334,196945057245451,1066104659977212

%N Number of multisets of exactly n nonempty words with a total of 2n letters over 2n-ary alphabet such that within each prefix of a word every letter of the alphabet is at least as frequent as the subsequent alphabet letter.

%H Alois P. Heinz, <a href="/A293809/b293809.txt">Table of n, a(n) for n = 0..800</a>

%F G.f.: Product_{j>=1} 1/(1-x^j)^A000085(j+1).

%F Euler transform of j-> A000085(j+1).

%F a(n) = A293808(2n,n).

%e a(0) = 1: {}.

%e a(1) = 2: {aa}, {ab}.

%e a(2) = 7: {a,aaa}, {a,aab}, {a,aba}, {a,abc}, {aa,aa}, {aa,ab}, {ab,ab}.

%p g:= proc(n) option remember; `if`(n<2, 1, g(n-1)+(n-1)*g(n-2)) end:

%p a:= proc(n) option remember; `if`(n=0, 1, add(add(d*

%p       g(d+1), d=numtheory[divisors](j))*a(n-j), j=1..n)/n)

%p     end:

%p seq(a(n), n=0..30);

%t g[n_] := g[n] = If[n < 2, 1, g[n-1] + (n-1)*g[n-2]];

%t a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*g[d+1], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n];

%t Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Nov 23 2023, after _Alois P. Heinz_ *)

%Y Cf. A000085, A293808.

%K nonn

%O 0,2

%A _Alois P. Heinz_, Oct 16 2017