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A294298 Sum of products of terms in all partitions of 3*n into powers of 3.

%I A294298 #39 Nov 23 2020 06:57:19
%S A294298 1,4,13,49,157,481,1534,4693,14170,43357,130918,393601,1188454,
%T A294298 3573013,10726690,32248957,96815758,290516161,872169223,2617128409,
%U A294298 7852005967,23561605318,70690403371,212076797530,636280680100,1908892327810,5726727270940,17180634420931
%N A294298 Sum of products of terms in all partitions of 3*n into powers of 3.
%H A294298 Seiichi Manyama, <a href="/A294298/b294298.txt">Table of n, a(n) for n = 0..1000</a>
%F A294298 a(n) = [x^(3*n)] Product_{k>=0} 1/(1 - 3^k*x^(3^k)). - _Ilya Gutkovskiy_, Sep 10 2018
%F A294298 a(n) ~ c * 3^n, where c = 2.2530906593645919365992433370928351696108819534655299832797806149219665... - _Vaclav Kotesovec_, Jun 18 2019
%e A294298 n | partitions of 3*n into powers of 3                          | a(n)
%e A294298 ----------------------------------------------------------------------------------
%e A294298 1 | 3  , 1+1+1                                                  | 3+1        =  4.
%e A294298 2 | 3+3, 3+1+1+1, 1+1+1+1+1+1                                   | 9+3+1      = 13.
%e A294298 3 | 9  , 3+3+3  , 3+3+1+1+1  , 3+1+1+1+1+1+1, 1+1+1+1+1+1+1+1+1 | 9+27+9+3+1 = 49.
%p A294298 b:= proc(n, i, p) option remember; `if`(n=0, p,
%p A294298      `if`(i<1, 0, add(b(n-j*i, i/3, p*i^j), j=0..n/i)))
%p A294298     end:
%p A294298 a:= n-> (t-> b(t, 3^ilog[3](t), 1))(3*n):
%p A294298 seq(a(n), n=0..33);  # _Alois P. Heinz_, Oct 27 2017
%t A294298 b[n_, i_, p_] := b[n, i, p] = If[n == 0, p, If[i < 1, 0, Sum[b[n - j i, i/3, p i^j], {j, 0, n/i}]]];
%t A294298 a[n_] := b[3n, 3^Floor@Log[3, 3n], 1];
%t A294298 a /@ Range[0, 33] (* _Jean-François Alcover_, Nov 23 2020, after _Alois P. Heinz_ *)
%Y A294298 Cf. A005704, A062051, A289842.
%K A294298 nonn
%O A294298 0,2
%A A294298 _Seiichi Manyama_, Oct 27 2017