A125801 -id:A125801 - cp's OEIS frontend

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

1, 2, 5, 23, 239, 5828, 342383, 50110484, 18757984046, 18318289003448, 47398244089264547, 329030840161393127681, 6190927493941741957366100, 318447442589056401640929570896, 45106654667152833836835578059359839

Offset: 0

Paul D. Hanna, Nov 18 2002

nonn

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)

			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

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.

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.

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

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]

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

A125800 Rectangular table where column k equals row sums of matrix power A078122^k, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 23, 12, 4, 1, 1, 239, 93, 22, 5, 1, 1, 5828, 1632, 238, 35, 6, 1, 1, 342383, 68457, 5827, 485, 51, 7, 1, 1, 50110484, 7112055, 342382, 15200, 861, 70, 8, 1, 1, 18757984046, 1879090014, 50110483, 1144664, 32856, 1393, 92, 9, 1

Offset: 0

Paul D. Hanna, Dec 10 2006

Determinant of n X n upper left submatrix is 3^(n*(n-1)*(n-2)/6).
This table is related to partitions of numbers into powers of 3 (see A078122).
Triangle A078122 shifts left one column under matrix cube.
Column 1 is A078125, which equals row sums of A078122;
column 2 is A078124, which equals row sums of A078122^2.

			Recurrence T(n,k) = T(n,k-1) + T(n-1,3*k) is illustrated by:
  T(3,3) = T(3,2) + T(2,9) = 93 + 145 = 238;
  T(4,3) = T(4,2) + T(3,9) = 1632 + 4195 = 5827;
  T(5,3) = T(5,2) + T(4,9) = 68457 + 273925 = 342382.
Rows of this table begin:
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...;
  1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ...;
  1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, ...;
  1, 23, 93, 238, 485, 861, 1393, 2108, 3033, 4195, 5621, ...;
  1, 239, 1632, 5827, 15200, 32856, 62629, 109082, 177507, 273925,...;
  1, 5828, 68457, 342382, 1144664, 3013980, 6769672, 13570796, ...;
  1, 342383, 7112055, 50110483, 215155493, 690729981, 1828979530, ...;
  1, 50110484, 1879090014, 18757984045, 103674882878, 406279238154,..;
  1, 18757984046, 1287814075131, 18318289003447, 130648799730635, ...;
Triangle A078122 begins:
  1;
  1,     1;
  1,     3,      1;
  1,    12,      9,     1;
  1,    93,    117,    27,    1;
  1,  1632,   3033,  1080,   81,   1;
  1, 68457, 177507, 86373, 9801, 243, 1; ...
where row sums form column 1 of this table A125790,
and column k of A078122 equals column 3^k - 1 of this table A125800.
Matrix square A078122^2 begins:
     1;
     2,     1;
     5,     6,     1;
    23,    51,    18,    1;
   239,   861,   477,   54,   1;
  5828, 32856, 25263, 4347, 162, 1; ...
where row sums form column 2 of this table A125790,
and column 0 of A078122^2 forms column 1 of this table A125790.

Robert Israel, Table of n, a(n) for n = 0..2484 (antidiagonals 0 to 69, flattened)

Cf. A078122; columns: A078125, A078124, A125801, A125802, A125803; A125804 (diagonal), A125805 (antidiagonal sums); related table: A125800 (q=2).

f[0]:= 1/(1-z):
S[0]:= series(f[0],z,21):
for n from 1 to 20 do
  ff:= unapply(f[n-1],z);
  f[n]:= simplify(1/3*sum(ff(w*z^(1/3)),w=RootOf(Z^3-1,Z)))/(1-z);
  S[n]:= series(f[n],z,21-n)
od:
seq(seq(coeff(S[s-i],z,i),i=0..s),s=0..20); # Robert Israel, Jun 02 2019

T[0, ] = T[, 0] = 1; T[n_, k_] := T[n, k] = T[n, k-1] + T[n-1, 3 k]; Table[T[n-k, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 08 2016 *)

T(n,k,p=0,q=3)=local(A=Mat(1), B); if(n

T(n,k) = T(n,k-1) + T(n-1,3*k) for n > 0, k > 0, with T(0,n)=T(n,0)=1 for n >= 0.

G.f. of row n is g_n(z) where g_{n+1}(z) = (1-z)^(-1)*Sum_{w^3=1} g_n(w*z^(1/3)) (the sum being over the cube roots of unity). - Robert Israel, Jun 02 2019

A125802 Column 4 of table A125800; also equals row sums of matrix power A078122^4.

Original entry on oeis.org

1, 5, 35, 485, 15200, 1144664, 215155493, 103674882878, 130648799730635, 437302448840089232, 3936208033244539574405, 96244898501021613327012635, 6446494058446469307795159512465, 1191218783863555524342034469450207222

Offset: 0

Paul D. Hanna, Dec 10 2006

nonn

Triangle A078122 shifts left one column under matrix cube and is related to partitions into powers of 3.

Cf. A125800, A078122; other columns: A078125, A078124, A125801, A125803.

a(n)=local(p=4,q=3,A=Mat(1), B); for(m=1, n+1, B=matrix(m, m); for(i=1, m, for(j=1, i, if(j==i || j==1, B[i, j]=1, B[i, j]=(A^q)[i-1, j-1]); )); A=B); return(sum(c=0,n,(A^p)[n+1,c+1]))

A125803 Column 5 of table A125800; also equals row sums of matrix power A078122^5.

Original entry on oeis.org

1, 6, 51, 861, 32856, 3013980, 690729981, 406279238154, 625750288074015, 2563196032703643450, 28270494794022487841733, 848050124165724284639262951, 69769378541879435090796205851249

Offset: 0

Paul D. Hanna, Dec 10 2006

nonn

Triangle A078122 shifts left one column under matrix cube and is related to partitions into powers of 3.

Cf. A125800, A078122; other columns: A078125, A078124, A125801, A125802.

a(n)=local(p=5,q=3,A=Mat(1), B); for(m=1, n+1, B=matrix(m, m); for(i=1, m, for(j=1, i, if(j==i || j==1, B[i, j]=1, B[i, j]=(A^q)[i-1, j-1]); )); A=B); return(sum(c=0,n,(A^p)[n+1,c+1]))

A125804 Main diagonal of table A125800.

Original entry on oeis.org

1, 2, 12, 238, 15200, 3013980, 1828979530, 3373190565626, 18837339867421686, 317817051628161116674, 16176220447967300610844988, 2481251352301850541661479580329, 1146112129196402690505198891390847384

Offset: 0

Paul D. Hanna, Dec 10 2006

nonn

Table A125800 is related to partitions into powers of 3; column k of A125800 equals row sums of matrix power A078122^k, where triangle A078122 shifts left one column under matrix cube.

Cf. A125800, A078122; columns: A078125, A078124, A125801, A125802, A125803; A125805 (antidiagonal sums).

a(n)=local(q=3,A=Mat(1), B); for(m=1, n+1, B=matrix(m, m); for(i=1, m, for(j=1, i, if(j==i || j==1, B[i, j]=1, B[i, j]=(A^q)[i-1, j-1]); )); A=B); return(sum(c=0,n,(A^n)[n+1,c+1]))

A125805 Antidiagonal sums of table A125800.

Original entry on oeis.org

1, 2, 4, 10, 41, 361, 7741, 417212, 57581062, 20688363559, 19625079296963, 49742424992663959, 340292157995636104240, 6337196928437059669994069, 323627960380394115802942263514, 45610724032832026072070666274435391

Offset: 0

Paul D. Hanna, Dec 10 2006

nonn

Table A125800 is related to partitions into powers of 3; column k of A125800 equals row sums of matrix power A078122^k, where triangle A078122 shifts left one column under matrix cube.

Cf. A125800, A078122; columns: A078125, A078124, A125801, A125802, A125803; A125804 (diagonal).

a(n)=local(q=3,A=Mat(1), B); for(m=1, n+1, B=matrix(m, m); for(i=1, m, for(j=1, i, if(j==i || j==1, B[i, j]=1, B[i, j]=(A^q)[i-1, j-1]); )); A=B); return(sum(c=0,n,(A^(c+1))[n-c+1,1]))

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

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A125800 Rectangular table where column k equals row sums of matrix power A078122^k, read by antidiagonals.

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A125802 Column 4 of table A125800; also equals row sums of matrix power A078122^4.

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A125803 Column 5 of table A125800; also equals row sums of matrix power A078122^5.

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A125804 Main diagonal of table A125800.

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A125805 Antidiagonal sums of table A125800.

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