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

1, 4, 13, 49, 157, 481, 1534, 4693, 14170, 43357, 130918, 393601, 1188454, 3573013, 10726690, 32248957, 96815758, 290516161, 872169223, 2617128409, 7852005967, 23561605318, 70690403371, 212076797530, 636280680100, 1908892327810, 5726727270940, 17180634420931

Offset: 0

Author

Seiichi Manyama, Oct 27 2017

Keywords

• nonn

Examples

			n | partitions of 3*n into powers of 3                          | a(n)
----------------------------------------------------------------------------------
1 | 3  , 1+1+1                                                  | 3+1        =  4.
2 | 3+3, 3+1+1+1, 1+1+1+1+1+1                                   | 9+3+1      = 13.
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.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A005704, A062051, A289842.

Programs

Maple

b:= proc(n, i, p) option remember; `if`(n=0, p,
     `if`(i<1, 0, add(b(n-j*i, i/3, p*i^j), j=0..n/i)))
    end:
a:= n-> (t-> b(t, 3^ilog[3](t), 1))(3*n):
seq(a(n), n=0..33);  # Alois P. Heinz, Oct 27 2017

Mathematica

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}]]];
a[n_] := b[3n, 3^Floor@Log[3, 3n], 1];
a /@ Range[0, 33] (* Jean-François Alcover, Nov 23 2020, after Alois P. Heinz *)

Formula

a(n) = [x^(3*n)] Product_{k>=0} 1/(1 - 3^k*x^(3^k)). - Ilya Gutkovskiy, Sep 10 2018

a(n) ~ c * 3^n, where c = 2.2530906593645919365992433370928351696108819534655299832797806149219665... - Vaclav Kotesovec, Jun 18 2019