0,2

a(n) = sum of the n-th row of lower triangular matrix of A078122. Conjecture: a(n) = the partitions of 3^n into powers of 3.

Number of partitions of 3^n into powers of 3. [From Valentin Bakoev (v_bakoev(AT)yahoo.com), Feb 22 2009]

Bakoev V., Algorithmic approach to counting of certain types m-ary partitions, Discrete Mathematics, 275 (2004) pp.17-41. [From Valentin Bakoev (v_bakoev(AT)yahoo.com), Feb 22 2009]

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 is: t_m(n, k)= k+1 if n=1, t_m(n, k)= 1 if k=0, and t_m(n, k)= t_m(n, k-1) + t_m(n-1, k.m) if n>1 and k>0. A078125 is obtained for m=3 and n=1,2,3,... [From Valentin Bakoev (v_bakoev(AT)yahoo.com), Feb 22 2009]

Square of A078122 = A078123 as can be seen by 4 X 4 submatrix:

[1,_0,_0,0]^2=[_1,_0,_0,_0]

[1,_1,_0,0]___[_2,_1,_0,_0]

[1,_3,_1,0]___[_5,_6,_1,_0]

[1,12,_9,1]___[23,51,18,_1]

To obtain t_3(5,2) 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,...,162(= k.m^{n-1}). It is: 1,2,3,4,5,6,7,8,...,162; 1,5,12,22,35,51,...,4510; (This row contains the first 55 members of A000326 - the pentagonal numbers) 1,23,93,238,485,...,29773; 1,239,1632,5827,15200,32856,62629; 1,5828,68457; Column 1 contains the first 5 members of A078125. [From Valentin Bakoev (v_bakoev(AT)yahoo.com), Feb 22 2009]

There exists an algorithm (with polynomial running-time) for computing the members of A078125, A125801 and many other sequences of the same type. [From Valentin Bakoev (v_bakoev(AT)yahoo.com), Feb 22 2009]

m[i_, j_] := m[i, j]=If[j==0||i==j, 1, m3[i-1, j-1]]; m2[i_, j_] := m2[i, j]=Sum[m[i, k]m[k, j], {k, j, i}]; m3[i_, j_] := m3[i, j]=Sum[m[i, k]m2[k, j], {k, j, i}]; a[n_] := m2[n, 0]

Cf. A078121, A078122 (matrix shift when cubed), A078123, A078124.

1) Subtracting 1 from the members of A078125 we obtain A125801. 2) 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. [From Valentin Bakoev (v_bakoev(AT)yahoo.com), Feb 22 2009]

Sequence in context: A136731 A062495 A158889 this_sequence A034692 A002507 A137094

Adjacent sequences: A078122 A078123 A078124 this_sequence A078126 A078127 A078128

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Nov 18 2002