A184714 - OEIS

A184714

Number of words of numbers x(1), ..., x(n), in 0..3 such that Sum_{i=1..n} i*x(i)^3 = 27*n.

%I #19 May 23 2023 07:28:10

%S 1,1,2,2,7,19,39,120,411,1310,3862,11059,31849,89615,247305,674579,

%T 1812940,4806211,12576930,32509641,83052304,209808493,524424707,

%U 1297623584,3179903180,7721053410,18582920108,44349211490,104989527861,246624184465,575024282279

%N Number of words of numbers x(1), ..., x(n), in 0..3 such that Sum_{i=1..n} i*x(i)^3 = 27*n.

%C Coefficient of x^(27*n) in Product_{i=1..n} (1 + x^i + x^(8*i) + x^(27*i)). - _Robert Israel_, Jul 03 2018

%H R. H. Hardin, <a href="/A184714/b184714.txt">Table of n, a(n) for n = 1..102</a>

%e All solutions for n=5:

%e 0 0 3 2 1 1 2

%e 0 3 0 2 1 2 1

%e 0 3 0 1 2 3 3

%e 0 0 3 3 3 2 1

%e 3 0 0 0 0 1 2

%p F:= proc(n,s) option remember; if n = 1 or s < 0 then 0 else add(procname(n-1,s-n*j^3),j=0..3) fi end proc:

%p F(1,0):= 1: F(1,1):= 1: F(1,8):= 1: F(1,27):= 1:

%p seq(F(n,n*27), n=1..50); # _Robert Israel_, Jul 03 2018

%t F[n_, s_] := F[n, s] = If[n == 1 || s < 0, 0, Sum[F[n-1, s-n*j^3], {j, 0, 3}]]; F[1, 0] = 1; F[1, 1] = 1; F[1, 8] = 1; F[1, 27] = 1;

%t Table[F[n, 27 n], {n, 1, 50}] (* _Jean-François Alcover_, May 23 2023, after _Robert Israel_ *)

%Y Column 3 of A184720.

%K nonn

%O 1,3

%A _R. H. Hardin_, Jan 20 2011