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A078125 Number of partitions of 3^n into powers of 3. 19

%I

%S 1,2,5,23,239,5828,342383,50110484,18757984046,18318289003448,

%T 47398244089264547,329030840161393127681,6190927493941741957366100,

%U 318447442589056401640929570896,45106654667152833836835578059359839

%N Number of partitions of 3^n into powers of 3.

%C a(n) = sum of the n-th row of lower triangular matrix of A078122.

%C From _Valentin Bakoev_, Feb 22 2009: (Start)

%C a(n) = the partitions of 3^n into powers of 3.

%C A125801(n) = a(n+1) - 1.

%C 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 Alois P. Heinz, <a href="/A078125/b078125.txt">Table of n, a(n) for n = 0..40</a>

%H 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 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 a(n) = [x^(3^n)] 1/Product_{j>=0} (1-x^(3^j)). - _Alois P. Heinz_, Sep 27 2011

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

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

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

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

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

%e 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 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 (Haskell)

%o import Data.MemoCombinators (memo2, list, integral)

%o a078125 n = a078125_list !! n

%o a078125_list = f [1] where

%o f xs = (p' xs $ last xs) : f (1 : map (* 3) xs)

%o p' = memo2 (list integral) integral p

%o p _ 0 = 1; p [] _ = 0

%o p ks'@(k:ks) m = if m < k then 0 else p' ks' (m - k) + p' ks m

%o -- _Reinhard Zumkeller_, Nov 27 2015

%Y Cf. A078121, A078122 (matrix shift when cubed), A078123, A078124, A125801.

%Y Column k=3 of A145515. - _Alois P. Heinz_, Sep 27 2011

%Y Cf. A000244, A002577, A145513.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Nov 18 2002

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Last modified June 25 10:28 EDT 2021. Contains 345453 sequences. (Running on oeis4.)