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

0,2

Links

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

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

Example

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.

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 *)

Crossrefs

Cf. A005704, A062051, A289842.

Sequence in context: A149449 A149450 A297591 * A300579 A180007 A338862

Adjacent sequences: A294295 A294296 A294297 * A294299 A294300 A294301

Keyword

nonn

Author

Seiichi Manyama, Oct 27 2017

Status

approved