login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified September 21 15:49 EDT 2019. Contains 327270 sequences. (Running on oeis4.)