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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 (list; graph; refs; listen; history; text; internal format)



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 (v_bakoev(AT)yahoo.com), Feb 20 2009]


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.


Denote the sum: m^n+m^n+...+m^n, k times, by k.m^n (m>1, n>0 and k are natural numbers). 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 (v_bakoev(AT)yahoo.com), Feb 20 2009]

From Valentin Bakoev (v_bakoev(AT)yahoo.com), Feb 20 2009: (Start)

Adding 1 to the members of A125801 we obtain A078125.

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 III, IV, etc. rows of the given table are not represented in OEIS till now. (End)

a(n) = A145515(n+1,3)-1. - Alois P. Heinz, Feb 27 2009


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 members of A125801. [Valentin Bakoev (v_bakoev(AT)yahoo.com), Feb 20 2009]


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[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-Fran├žois Alcover, Jan 21 2017 *)


(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]))


Cf. A125800, A078122; other columns: A078125, A078124, A125802, A125803.

Sequence in context: A260296 A137158 A025135 * A195227 A265908 A119009

Adjacent sequences:  A125798 A125799 A125800 * A125802 A125803 A125804




Paul D. Hanna, Dec 10 2006



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Last modified October 23 22:26 EDT 2017. Contains 293833 sequences.