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%I #26 Feb 23 2019 04:23:57
%S 1,4,22,238,5827,342382,50110483,18757984045,18318289003447,
%T 47398244089264546,329030840161393127680,6190927493941741957366099,
%U 318447442589056401640929570895,45106654667152833836835578059359838
%N Column 3 of table A125800; also equals row sums of matrix power A078122^3.
%C Triangle A078122 shifts left one column under matrix cube and is related to partitions into powers of 3.
%C Number of partitions of 3^n into powers of 3, excluding the trivial partition 3^n=3^n. - _Valentin Bakoev_, Feb 20 2009
%H Alois P. Heinz, <a href="/A125801/b125801.txt">Table of n, a(n) for n = 0..40</a>
%H V. Bakoev, <a href="http://dx.doi.org/10.1016/S0012-365X(03)00096-7">Algorithmic approach to counting of certain types m-ary partitions</a>, Discrete Mathematics, 275 (2004) pp. 17-41.
%F 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
%F From _Valentin Bakoev_, Feb 20 2009: (Start)
%F Adding 1 to the terms of A125801 we obtain A078125.
%F 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)
%F a(n) = A145515(n+1,3)-1. - _Alois P. Heinz_, Feb 27 2009
%e 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
%p 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
%t T[0, _] = T[_, 0] = 1; T[n_, k_] := T[n, k] = T[n, k-1] + T[n-1, 3 k];
%t a[n_] := T[n, 3]; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 15}] (* _Jean-François Alcover_, Jan 21 2017 *)
%o (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]))
%Y Cf. A125800, A078122; other columns: A078125, A078124, A125802, A125803.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Dec 10 2006
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