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%I A109055 #17 Apr 01 2024 13:06:51
%S A109055 1,1,3,24,541,35649,6979689,4085743032,7166723910237,
%T A109055 37698139930450365,594816080266215640710,28154472624850002001979592,
%U A109055 3997853576535778666975681355079,1703042427700923785323670557504832751,2176429411666209822350337722381643148477248
%N A109055 To compute a(n) we first write down 3^n 1's in a row. Each row takes the rightmost 3rd part of the previous row and each element in it equals sum of the elements of the previous row starting with the first of the rightmost 3rd part. The single element in the last row is a(n).
%C A109055 Comment from _Franklin T. Adams-Watters_, Jul 13 2006: This is the number of subpartitions of the sequence 3^n-1. As such it can also be computed adding forward, with 3^n terms in the n-th line:
%C A109055 1...........................................................................
%C A109055 1.1 1.......................................................................
%C A109055 1.2.3.3..3..3..3..3..3......................................................
%C A109055 1.3.6.9.12.15.18.21.24.24.24.24.24.24.24.24.24.24.24.24.24.24.24.24.24.24.24
%H A109055 Alois P. Heinz, <a href="/A109055/b109055.txt">Table of n, a(n) for n = 0..65</a>
%e A109055 For example, for n=3 the array looks like this:
%e A109055 1..1..1..1..1........1..1..1..1..1..1..1..1..1..1
%e A109055 ........................1..2..3..4..5..6..7..8..9
%e A109055 ..........................................7.15.24
%e A109055 ...............................................24
%e A109055 Therefore a(3)=24.
%p A109055 proc(n::nonnegint) local f,a; if n=0 or n=1 then return 1; end if; f:=L->[seq(add(L[i],i=2*nops(L)/3+1..j),j=2*nops(L)/3+1..nops(L))]; a:=f([seq(1,j=1..3^n)]); while nops(a)>3 do a:=f(a) end do; a[3]; end proc;
%t A109055 A[n_, k_] := A[n, k] = If[n == 0, 1, -Sum[A[j, k]*(-1)^(n - j)*Binomial[If[j == 0, 1, k^j], n - j], {j, 0, n - 1}]];
%t A109055 a[n_] := A[n, 3];
%t A109055 Table[a[n], {n, 0, 14}] (* _Jean-François Alcover_, Apr 01 2024, after _Alois P. Heinz_ in A355576 *)
%Y A109055 Cf. A107354, A109056, A109057, A109058, A109059, A109060, A109061, A109062.
%Y A109055 Cf. A115728, A115729.
%Y A109055 Column k=3 of A355576.
%K A109055 nonn
%O A109055 0,3
%A A109055 _Augustine O. Munagi_, Jun 17 2005
%E A109055 More terms from _Paul D. Hanna_

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