

A125801


Column 3 of table A125800; also equals row sums of matrix power A078122^3.


7



1, 4, 22, 238, 5827, 342382, 50110483, 18757984045, 18318289003447, 47398244089264546, 329030840161393127680, 6190927493941741957366099, 318447442589056401640929570895, 45106654667152833836835578059359838
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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



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, k1) + Sum_{j=1..m} t_m(n1, (k1)*n+j)}, when n > 1 and k > 0. A125801 is obtained for m=3 and n=1,2,3,...  Valentin Bakoev, Feb 20 2009
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)


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^{n1}). 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(bt, n, k) *binomial (n+1, t) *(1)^(t+1), t=1..n+1); else g(b1, n, k) +g(b*k, n1, 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, k1] + T[n1, 3 k];
a[n_] := T[n, 3]; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 15}] (* JeanFrançois 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)[i1, j1]); )); A=B); return(sum(c=0, n, (A^p)[n+1, c+1]))


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



