cp's OEIS Frontend

A078125 Number of partitions of 3^n into powers of 3.

%I A078125 #40 Feb 23 2019 19:49:46
%S A078125 1,2,5,23,239,5828,342383,50110484,18757984046,18318289003448,
%T A078125 47398244089264547,329030840161393127681,6190927493941741957366100,
%U A078125 318447442589056401640929570896,45106654667152833836835578059359839
%N A078125 Number of partitions of 3^n into powers of 3.
%C A078125 a(n) = sum of the n-th row of lower triangular matrix of A078122.
%C A078125 From _Valentin Bakoev_, Feb 22 2009: (Start)
%C A078125 a(n) = the partitions of 3^n into powers of 3.
%C A078125 A125801(n) = a(n+1) - 1.
%C 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 3rd, 4th, etc. rows of the given table are not represented in the OEIS till now. (End)
%H A078125 Alois P. Heinz, <a href="/A078125/b078125.txt">Table of n, a(n) for n = 0..40</a>
%H A078125 V. Bakoev, <a href="https://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 A078125 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. a(n) is obtained for m=3 and n=1,2,3,... - _Valentin Bakoev_, Feb 22 2009
%F A078125 a(n) = [x^(3^n)] 1/Product_{j>=0} (1-x^(3^j)). - _Alois P. Heinz_, Sep 27 2011
%e A078125 Square of A078122 = A078123 as can be seen by 4 X 4 submatrix:
%e A078125 [1,_0,_0,0]^2=[_1,_0,_0,_0]
%e A078125 [1,_1,_0,0]___[_2,_1,_0,_0]
%e A078125 [1,_3,_1,0]___[_5,_6,_1,_0]
%e A078125 [1,12,_9,1]___[23,51,18,_1]
%e A078125 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 this sequence. - _Valentin Bakoev_, Feb 22 2009
%t A078125 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]
%o A078125 (Haskell)
%o A078125 import Data.MemoCombinators (memo2, list, integral)
%o A078125 a078125 n = a078125_list !! n
%o A078125 a078125_list = f [1] where
%o A078125    f xs = (p' xs $ last xs) : f (1 : map (* 3) xs)
%o A078125    p' = memo2 (list integral) integral p
%o A078125    p _ 0 = 1; p [] _ = 0
%o A078125    p ks'@(k:ks) m = if m < k then 0 else p' ks' (m - k) + p' ks m
%o A078125 -- _Reinhard Zumkeller_, Nov 27 2015
%Y A078125 Cf. A078121, A078122 (matrix shift when cubed), A078123, A078124, A125801.
%Y A078125 Column k=3 of A145515. - _Alois P. Heinz_, Sep 27 2011
%Y A078125 Cf. A000244, A002577, A145513.
%K A078125 nonn
%O A078125 0,2
%A A078125 _Paul D. Hanna_, Nov 18 2002