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A078125 First column of matrix A078123, which is the square of the infinite lower triangular matrix A078122 that shifts left and up when cubed. 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. Conjecture: a(n) = the partitions of 3^n into powers of 3.

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

REFERENCES

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

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. A078125 is obtained for m=3 and n=1,2,3,... - Valentin Bakoev (v_bakoev(AT)yahoo.com), 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 A078125. - Valentin Bakoev (v_bakoev(AT)yahoo.com), 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.

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. - Valentin Bakoev (v_bakoev(AT)yahoo.com), Feb 22 2009

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 29 10:48 EDT 2017. Contains 287246 sequences.