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 A078125 Number of partitions of 3^n into powers of 3. 19
 1, 2, 5, 23, 239, 5828, 342383, 50110484, 18757984046, 18318289003448, 47398244089264547, 329030840161393127681, 6190927493941741957366100, 318447442589056401640929570896, 45106654667152833836835578059359839 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS a(n) = sum of the n-th row of lower triangular matrix of A078122. From Valentin Bakoev, Feb 22 2009: (Start) a(n) = the partitions of 3^n into powers of 3. A125801(n) = a(n+1) - 1. 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) LINKS 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. FORMULA 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 a(n) = [x^(3^n)] 1/Product_{j>=0} (1-x^(3^j)). - Alois P. Heinz, Sep 27 2011 EXAMPLE 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 this sequence. - Valentin Bakoev, Feb 22 2009 MATHEMATICA 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] PROG (Haskell) import Data.MemoCombinators (memo2, list, integral) a078125 n = a078125_list !! n a078125_list = f [1] where    f xs = (p' xs \$ last xs) : f (1 : map (* 3) xs)    p' = memo2 (list integral) integral p    p _ 0 = 1; p [] _ = 0    p ks'@(k:ks) m = if m < k then 0 else p' ks' (m - k) + p' ks m -- Reinhard Zumkeller, Nov 27 2015 CROSSREFS Cf. A078121, A078122 (matrix shift when cubed), A078123, A078124, A125801. Column k=3 of A145515. - Alois P. Heinz, Sep 27 2011 Cf. A000244, A002577, A145513. Sequence in context: A062495 A158889 A181074 * A034692 A002507 A137094 Adjacent sequences:  A078122 A078123 A078124 * A078126 A078127 A078128 KEYWORD nonn AUTHOR Paul D. Hanna, Nov 18 2002 STATUS approved

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Last modified May 9 12:38 EDT 2021. Contains 343740 sequences. (Running on oeis4.)
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