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%I A292838 #10 Dec 30 2020 08:22:27
%S A292838 1,4,22,132,729,4000,21488,113760,594548,3073392,15732936,79846448,
%T A292838 402104884,2010879968,9992425872,49366096352,242584319710,
%U A292838 1186177166680,5773569726884,27982357252632,135079969593838,649640609539360,3113354757088720,14871179093155424
%N A292838 Number of sets of nonempty words with a total of n letters over quaternary alphabet.
%H A292838 Alois P. Heinz, <a href="/A292838/b292838.txt">Table of n, a(n) for n = 0..1000</a>
%F A292838 G.f.: Product_{j>=1} (1+x^j)^(4^j).
%F A292838 a(n) ~ 4^n * exp(2*sqrt(n) - 1/2 - c) / (2 * sqrt(Pi) * n^(3/4)), where c = Sum_{m>=2} (-1)^m/(m*(4^(m-1)-1)) = 0.147762663788961720137665013823002812172... - _Vaclav Kotesovec_, Sep 28 2017
%p A292838 h:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A292838       add(h(n-i*j, i-1)*binomial(4^i, j), j=0..n/i)))
%p A292838     end:
%p A292838 a:= n-> h(n$2):
%p A292838 seq(a(n), n=0..30);
%t A292838 h[n_, i_] := h[n, i] = If[n == 0, 1, If[i < 1, 0,
%t A292838     Sum[h[n - i j,  i - 1] Binomial[4^i, j], {j, 0, n/i}]]];
%t A292838 a[n_] := h[n, n];
%t A292838 a /@ Range[0, 30] (* _Jean-François Alcover_, Dec 30 2020, after _Alois P. Heinz_ *)
%Y A292838 Column k=4 of A292804.
%K A292838 nonn
%O A292838 0,2
%A A292838 _Alois P. Heinz_, Sep 24 2017

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