A293737 - OEIS

%I #9 May 30 2019 09:27:55

%S 1,1,3,7,20,54,164,500,1629,5462,19164,69457,261154,1012164,4045640,

%T 16611121,70001515,301922104,1331128134,5986321599,27426419974,

%U 127801386949,605016657100,2906093083727,14149469612919,69762426194708,348016146152252,1755188873640756

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

%C This sequence differs from A293110 first at n=8.

%H Alois P. Heinz, <a href="/A293737/b293737.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: Product_{j>=1} 1/(1-x^j)^A007578(j).

%F a(n) ~ c * 7^n / n^(21/2), where c = 233774941.39802934196800791705821024006230754487492494942398064537776753785... - _Vaclav Kotesovec_, May 30 2019

%p g:= proc(n) option remember; `if`(n<4, [1, 1, 2, 4][n+1],

%p       ((4*n^3+78*n^2+424*n+495)*g(n-1) +(n-1)*(34*n^2+280*n+

%p        305)*g(n-2) -2*(n-1)*(n-2)*(38*n+145)*g(n-3) -105*(n-1)

%p        *(n-2)*(n-3)*g(n-4))/((n+6)*(n+10)*(n+12)))

%p     end:

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

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

%p     end:

%p seq(a(n), n=0..35);

%Y Column k=7 of A293108.

%Y Cf. A007578, A293110, A293746.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Oct 15 2017