OFFSET
0,2
COMMENTS
Triangle A078122 shifts left one column under matrix cube and is related to partitions into powers of 3.
Number of partitions of 3^n into powers of 3, excluding the trivial partition 3^n=3^n. - Valentin Bakoev, Feb 20 2009
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..40
V. Bakoev, Algorithmic approach to counting of certain types m-ary partitions, Discrete Mathematics, 275 (2004) pp. 17-41.
FORMULA
Denote the sum: m^n +m^n + ... + m^n, k times, by k*m^n (m > 1, n > 0 and k are positive integers). The general formula for the number of all partitions of the sum k*m^n into powers of m smaller than m^n, is t_m(n, k)= 1 when n=1 or k=0, or = t_m(n, k-1) + Sum_{j=1..m} t_m(n-1, (k-1)*n+j)}, when n > 1 and k > 0. A125801 is obtained for m=3 and n=1,2,3,... - Valentin Bakoev, Feb 20 2009
From Valentin Bakoev, Feb 20 2009: (Start)
For given m, the general formula for t_m(n, k) and the corresponding tables T, computed as in the example, determine a family of related sequences (placed in the rows or in the columns of T). For example, the sequences from the 3rd, 4th, etc. rows of the given table are not represented in the OEIS till now. (End)
a(n) = A145515(n+1,3)-1. - Alois P. Heinz, Feb 27 2009
EXAMPLE
To obtain t_3(5,1) we use the table T, defined as T(i,j) = t_3(i,j), for i=1,2,...,5(=n), and j=0,1,2,...,81(= k*m^{n-1}). It is 1,1,1,1,1,1,...1; 1,4,7,10,13,...,82; 1,22,70,145,247,376,532,715,925,1162; 1,238,1393,4195; 1,5827; Column 1 contains the first 5 terms of A125801. - Valentin Bakoev, Feb 20 2009
MAPLE
g:= proc(b, n, k) option remember; local t; if b<0 then 0 elif b=0 or n=0 or k<=1 then 1 elif b>=n then add (g(b-t, n, k) *binomial (n+1, t) *(-1)^(t+1), t=1..n+1); else g(b-1, n, k) +g(b*k, n-1, k) fi end: a:= n-> g(1, n+1, 3)-1: seq(a(n), n=0..25); # Alois P. Heinz, Feb 27 2009
MATHEMATICA
T[0, _] = T[_, 0] = 1; T[n_, k_] := T[n, k] = T[n, k-1] + T[n-1, 3 k];
a[n_] := T[n, 3]; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 15}] (* Jean-François Alcover, Jan 21 2017 *)
PROG
(PARI) a(n)=local(p=3, q=3, A=Mat(1), B); for(m=1, n+1, B=matrix(m, m); for(i=1, m, for(j=1, i, if(j==i || j==1, B[i, j]=1, B[i, j]=(A^q)[i-1, j-1]); )); A=B); return(sum(c=0, n, (A^p)[n+1, c+1]))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 10 2006
STATUS
approved