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A125801 Column 3 of table A125800; also equals row sums of matrix power A078122^3.

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