A002577 -id:A002577 - 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

A125790 Rectangular table where column k equals row sums of matrix power A078121^k, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 10, 9, 4, 1, 1, 36, 35, 16, 5, 1, 1, 202, 201, 84, 25, 6, 1, 1, 1828, 1827, 656, 165, 36, 7, 1, 1, 27338, 27337, 8148, 1625, 286, 49, 8, 1, 1, 692004, 692003, 167568, 25509, 3396, 455, 64, 9, 1, 1, 30251722, 30251721, 5866452, 664665, 64350, 6321, 680, 81, 10, 1

Offset: 0

Paul D. Hanna, Dec 10 2006, corrected Dec 12 2006

Determinant of n X n upper left submatrix is 2^[n(n-1)(n-2)/6] (see A125791). Related to partitions of numbers into powers of 2 (see A078121). Triangle A078121 shifts left one column under matrix square.

			Recurrence T(n,k) = T(n,k-1) + T(n-1,2*k) is illustrated by:
  T(4,3) = T(4,2) + T(3,6) = 201 + 455 = 656;
  T(5,3) = T(5,2) + T(4,6) = 1827 + 6321 = 8148;
  T(6,3) = T(6,2) + T(5,6) = 27337 + 140231 = 167568.
Rows of this table begin:
  1, 1, 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, 13, 14, ...;
  1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, ...;
  1, 10, 35, 84, 165, 286, 455, 680, 969, 1330, 1771, 2300, ...;
  1, 36, 201, 656, 1625, 3396, 6321, 10816, 17361, 26500, 38841, ...;
  1, 202, 1827, 8148, 25509, 64350, 140231, 274856, 497097, ...;
  1, 1828, 27337, 167568, 664665, 2026564, 5174449, 11622976, ...;
  1, 27338, 692003, 5866452, 29559717, 109082974, 326603719, ...;
  1, 692004, 30251721, 356855440, 2290267225, 10243585092, ...; ...
Triangle A078121 begins:
  1;
  1,   1;
  1,   2,   1;
  1,   4,   4,   1;
  1,  10,  16,   8,   1;
  1,  36,  84,  64,  16,  1;
  1, 202, 656, 680, 256, 32, 1; ...
where row sums form column 1 of this table A125790,
and column k of A078121 equals column 2^k-1 of this table A125790.
Matrix cube A078121^3 begins:
     1;
     3,    1;
     9,    6,    1;
    35,   36,   12,   1;
   201,  286,  144,  24,  1;
  1827, 3396, 2300, 576, 48, 1; ...
where row sums form column 3 of this table A125790,
and column 0 of A078121^3 forms column 2 of this table A125790.

G. Blom and C.-E. Froeberg, Om myntvaexling (On money-changing) [Swedish], Nordisk Matematisk Tidskrift, 10 (1962), 55-69, 103. [Annotated scanned copy] See Table 5.

Cf. A078121; A002577; A125791; columns: A002577, A125792, A125793, A125794, A125795, A125796; diagonals: A125797, A125798; A125799 (antidiagonal sums); related table: A125800 (q=3).

T[n_, k_] := T[n, k] = T[n, k-1] + T[n-1, 2*k]; T[0, ] = T[, 0] = 1; Table[T[n-k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 15 2015 *)

{T(n,k,p=0,q=2)=local(A=Mat(1), B); if(n

T(n,k) = T(n,k-1) + T(n-1,2*k) for n>0, k>0, with T(0,n)=T(n,0)=1 for n>=0.

Conjecture: g.f. for n-th row is (Sum_{i=0..n-1} x^i Sum_{j=0..i} binomial(n+1,j)*T(n,i-j)*(-1)^j)/(1-x)^(n+1) for n > 0. - Mikhail Kurkov, May 03 2025

A125791 a(n) = 2^(n(n-1)(n-2)/6) for n>=1.

Original entry on oeis.org

1, 1, 2, 16, 1024, 1048576, 34359738368, 72057594037927936, 19342813113834066795298816, 1329227995784915872903807060280344576, 46768052394588893382517914646921056628989841375232

Offset: 1

Paul D. Hanna, Dec 10 2006

nonn

a(n) is a tetrahedral power of 2; exponents of 2 in a(n) begin: 0, 0, 1, 4, 10, 20, 35, 56, 84, 120, 165, ..., n*(n-1)*(n-2)/6, ... (cf. A000292).
Table A125790 is related to partitions into powers of 2, with A002577 in column 1 of A125790; further, column k of A125790 equals row sums of matrix power A078121^k, where triangle A078121 shifts left one column under matrix square.
Also number of distinct instances of the one-in-three monotone 3SAT problem for n variables. - Paul Tarau (paul.tarau(AT)gmail.com), Jan 25 2008
Hankel transform of aerated 2-Catalan numbers (A015083). [Paul Barry, Dec 15 2010]

Michael De Vlieger, Table of n, a(n) for n = 1..28
Pakawut Jiradilok, Some Combinatorial Formulas Related to Diagonal Ramsey Numbers, arXiv:2404.02714 [math.CO], 2024. See p. 19.

Cf. A125790, A078121; A002577, A000292.

seq(2^(binomial(n, n-3)), n=1..10); # Zerinvary Lajos, Jun 16 2007 [modified by Georg Fischer, Nov 09 2023]

A125791[n_]:=2^Binomial[n,n-3];Array[A125791,15] (* Paolo Xausa, Nov 05 2023 *)

PARI
```
a(n)=if(n<1,0,2^(n*(n-1)*(n-2)/6))
```

/* As determinant of n X n matrix: */
{a(n)=local(q=2,A=Mat(1), B); for(m=1, n, 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(matdet(matrix(n,n,r,c,(A^c)[r,1])))}
for(n=1,15,print1(a(n),", "))

% This generates all 3SAT problem instances
test:-test(4).
test(Max):-
between(1,Max,N),
nl,
one_in_three_monotone_3sat(N,Pss),
write(N:Pss),nl,
fail
; nl.
% generates all one-in-three monotone 3SAT problems involving N variables
one_in_three_monotone_3sat(N,Pss):-
ints(1,N,Is),
findall(Xs,ksubset(3,Is,Xs),Xss),
subset_of(Xss,Pss).
% subset generator
subset_of([],[]).
subset_of([X|Xs],Zs):-
subset_of(Xs,Ys),
add_element(X,Ys,Zs).
add_element(_,Ys,Ys).
add_element(X,Ys,[X|Ys]).
% subsets of K elements
ksubset(0,_,[]).
ksubset(K,[X|Xs],[X|Rs]):-K>0,K1 is K-1,ksubset(K1,Xs,Rs).
ksubset(K,[_|Xs],Rs):-K>0,ksubset(K,Xs,Rs).
% list of integers in [From..To]
ints(From,To,Is):-findall(I,between(From,To,I),Is).
% Paul Tarau (paul.tarau(AT)gmail.com), Jan 25 2008

Determinant of n X n upper left corner submatrix of table A125790.

a(n) = 2^(binomial(n,n-3)). - Zerinvary Lajos, Jun 16 2007, modified to reflect the new offset by Paolo Xausa, Nov 06 2023.

Name simplified; determinant formula moved out of name into formula section by Paul D. Hanna, Oct 16 2013

Offset changed to 1 by Paolo Xausa, Nov 06 2023

A152977 Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) is the number of partitions of 2^n into powers of 2 less than or equal to 2^k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 4, 5, 1, 1, 2, 4, 9, 9, 1, 1, 2, 4, 10, 25, 17, 1, 1, 2, 4, 10, 35, 81, 33, 1, 1, 2, 4, 10, 36, 165, 289, 65, 1, 1, 2, 4, 10, 36, 201, 969, 1089, 129, 1, 1, 2, 4, 10, 36, 202, 1625, 6545, 4225, 257, 1, 1, 2, 4, 10, 36, 202, 1827, 17361, 47905, 16641, 513, 1

Offset: 0

Alois P. Heinz, Jan 26 2011

Column sequences converge towards A002577.

			A(3,2) = 9, because there are 9 partitions of 2^3=8 into powers of 2 less than or equal to 2^2=4: [4,4], [4,2,2], [4,2,1,1], [4,1,1,1,1], [2,2,2,2], [2,2,2,1,1], [2,2,1,1,1,1], [2,1,1,1,1,1,1], [1,1,1,1,1,1,1,1].
Square array A(n,k) begins:
  1,  1,  1,   1,   1,   1,  ...
  1,  2,  2,   2,   2,   2,  ...
  1,  3,  4,   4,   4,   4,  ...
  1,  5,  9,  10,  10,  10,  ...
  1,  9, 25,  35,  36,  36,  ...
  1, 17, 81, 165, 201, 202,  ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-10 give: A000012, A094373, A028400(n-2) for n>1, A210772, A210773, A210774, A210775, A210776, A210777, A210778, A210779.
Cf. A002577, A000123, A181322, A145515.
Main diagonal and lower diagonals give: A002577, A125792, A125794.

b:= proc(n,j) local nn, r;
      if n<0 then 0
    elif j=0 then 1
    elif j=1 then n+1
    elif n `if`(n=0, 1, b(2^(n-k), k)):
seq(seq(A(n, d-n), n=0..d), d=0..11);

b[n_, j_] := Module[{nn, r}, Which[n < 0, 0, j == 0, 1, j == 1, n+1, n < j, b[n, j] = b[n-1, j]+b[2*n, j-1], True, nn = 1+Floor[n]; r := n-nn; (nn-j)*Binomial[nn, j]*Sum[Binomial[j, h]/(nn-j+h)*b[j-h+r, j]*(-1)^h, {h, 0, j-1}]]]; a[n_, k_] := If[n == 0, 1, b[2^(n-k), k]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 11}] // Flatten (* Jean-François Alcover, Dec 18 2013, translated from Maple *)

A(n,k) = [x^2^(n-1)] 1/(1-x) * 1/Product_{j=0..k-1} (1-x^(2^j)) for n>0; A(0,k) = 1.

A125792 Column 2 of table A125790; also equals row sums of matrix power A078121^2.

Original entry on oeis.org

1, 3, 9, 35, 201, 1827, 27337, 692003, 30251721, 2320518947, 316359580361, 77477180493603, 34394869942983369, 27893897106768940835, 41603705003444309596873, 114788185359199234852802339, 588880400923055731115178072777, 5642645813427132737155703265972003

Offset: 0

Paul D. Hanna, Dec 10 2006

nonn

Triangle A078121 shifts left one column under matrix square and is related to partitions into powers of 2.

Number of partitions of 2^n into powers of 2, excluding the trivial partition 2^n=2^n. - Valentin Bakoev, Feb 15 2009

			G.f.: 1 + 3*x + 9*x^2 + 35*x^3 + 201*x^4 + 1827*x^5 + 27337*x^6 + 692003*x^7 + ...
To obtain t_2(5,1) we use the table T, defined as T[i,j]= t_2(i,j), for i=1,2,...,5(=n), and j= 0,1,2,...,16(= k*m^{n-1}). It is 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1 1,3,5,7,9,11,13,15,17 1,9,25,49,81 1,35,165 1,201 Column 1 contains the first 5 members of A125792. [_Valentin Bakoev_, Feb 15 2009]

Alois P. Heinz, Table of n, a(n) for n = 0..86
V. Bakoev, Algorithmic approach to counting of certain types m-ary partitions, Discrete Mathematics, 275 (2004) pp. 17-41.

Cf. A125790, A078121; A002575; columns: A002577, A125793, A125794, A125795, A125796; diagonals: A125797, A125798.
Adding 1 to the members of A125792 we obtain A002577. [Valentin Bakoev, Feb 15 2009]
A diagonal of A152977.

g:= proc(b, n, k) option remember; local t; if b<0 then 0 elif b=0 or n=0 or k<=1 then 1 elif b>=n then add(g(b-t, n, k) *binomial(n+1, t) *(-1)^(t+1), t=1..n+1); else g(b-1, n, k) +g(b*k, n-1, k) fi end: a:= n-> g(1, n+1,2)-1: seq(a(n), n=0..25);  # Alois P. Heinz, Feb 27 2009

T[n_, k_] := T[n, k] = T[n, k-1] + T[n-1, 2*k]; T[0, ] = T[, 0] = 1; Table[T[n, 2], {n, 0, 20} ] (* Jean-François Alcover, Jun 15 2015 *)

{a(n)=local(p=2,q=2,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]))}
for(n=0,25,print1(a(n),", "))

{a(n, k=3) = if(n<1, n==0, sum(i=1, k, a(n-1, 2*i-1)))}; /* Michael Somos, Nov 24 2016 */

Is this sequence the same as A002575 (coefficients of Bell's 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, smaller than m^n, is t_m(n, k)= 1 when n=1 or k=0, or = t_m(n, k-1) + Sum_{j=1..m} t_m(n-1, (k-1)*n+j), when n > 1 and k > 0. A125792 is obtained for m=2 and n=1,2,3,... [Valentin Bakoev, Feb 15 2009]
a(n) = A145515(n+1,2)-1. - Alois P. Heinz, Feb 27 2009
From Benedict W. J. Irwin, Nov 16 2016: (Start)
Conjecture: a(n+1) = Sum_{i_1=1..3} Sum_{i_2=1..2*i_1-1} ... Sum_{i_n=1..2*i_(n-1)-1} (2*i_n - 1). For example:
a(2) = Sum_{i=1..3} 2*i-1.
a(3) = Sum_{i=1..3} Sum_{j=1..2*i-1} 2*j-1.
a(4) = Sum_{i=1..3} Sum_{j=1..2*i-1} Sum_{k=1..2*j-1} 2*k-1. (End)

A078537 Number of partitions of 4^n into powers of 4 (without regard to order).

Original entry on oeis.org

1, 2, 6, 46, 1086, 79326, 18583582, 14481808030, 38559135542174, 357934565638890910, 11766678027350761752990, 1387043469046575118555443614, 592264246356176268834689653440926, 923812464024548700407122072128655860126, 5301247577915139769925461060755690116740047262

Offset: 0

Paul D. Hanna, Nov 29 2002

nonn

Conjecture: a(n) = sum of the n-th row of lower triangular matrix A078536.

			a(2) = 6 since partitions of 4^2 into powers of 4 are: [16], [4,4,4,4], [4,4,4,1,1,1,1], [4,4,1,1,1,1,1,1,1,1], [4,1,1,1,1,1,1,1,1,1,1,1,1], [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1].

Alois P. Heinz, Table of n, a(n) for n = 0..60

Cf. A002577, A078125, A078536.

Column k=4 of A145515.

a[0] = 1; a[n_] := a[n] = a[n - 1] + a[Floor[n/4]]; b = Table[ a[n], {n, 0, 4^9}]; Table[ b[[4^n + 1]], {n, 0, 9}]

a(n) = coefficient of x^(4^n) in power series expansion of 1/[(1-x)(1-x^4)(1-x^16)...(1-x^(4^k))...].

Extended by Robert G. Wilson v, Dec 01 2002

More terms from Alois P. Heinz, Oct 11 2008

A002575 Coefficients of Bell's formula for making change.

Original entry on oeis.org

1, 3, 9, 35, 201, 1827, 27337, 692003, 30251721, 2320518947, 316359580361, 77477180493603, 34394869942983369, 27893897106768940835, 41603705003444309596873, 114788185359199234852802339, 588880400923055731115178072777

Offset: 2

N. J. A. Sloane

nonn

G. Blom and C.-E. Froeberg, Om myntvaexling (On money-changing) [Swedish], Nordisk Matematisk Tidskrift, 10 (1962), 55-69, 103.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. Blom and C.-E. Froeberg, Om myntvaexling (On money-changing) [Swedish], Nordisk Matematisk Tidskrift, 10 (1962), 55-69, 103. [Annotated scanned copy]
G. Blom and C.-E. Froeberg, Om myntvaexling, (On money-changing) [Swedish], Nordisk Matematisk Tidskrift, 10 (1962), 55-69, 103.

Cf. A002576, A002577, A125792.

A diagonal of A262554.

a(n) = 2*a(n-1) + A002576(n-1) + 1. - Max Alekseyev, Mar 17 2023

More terms from Max Alekseyev, Mar 17 2023

A111814 Column 0 of the matrix logarithm (A111813) of triangle A078121, which shifts columns left and up under matrix square; these terms are the result of multiplying the element in row n by n!.

Original entry on oeis.org

0, 1, 0, -2, 0, 216, 0, -568464, 0, 36058658688, 0, -53694310935340800, 0, 1790669979087018171448320, 0, -1280832788659041410080025283840000, 0, 18961468161294510864200732026858464699187200, 0

Offset: 0

Gottfried Helms and Paul D. Hanna, Aug 22 2005

sign

Surprisingly, the e.g.f. A(x) is an odd function: A(-x) = -A(x). Let q=2; the g.f. of column k of A078121^m (matrix power m) is: 1 + Sum_{n>=1} (m*q^k)^n/n! * Product_{j=0..n-1} A(q^j*x).

			E.g.f.: A(x) = x - 2/3!*x^3 + 216/5!*x^5 - 568464/7!*x^7 + ...
where A(x) satisfies:
x/(1-x) = A(x) + A(x)*A(2*x)/2! + A(x)*A(2*x)*A(2^2*x)/3!
+ A(x)*A(2*x)*A(2^2*x)*A(2^3*x)/4! + ...
also:
x/(1-x^2) = A(x) + A(x)*A(2*x)*A(2^2*x)/3!
+ A(x)*A(2*x)*A(2^2*x)*A(2^3*x)*A(2^4*x)/5! + ...
Let G(x) be the g.f. of A002577 (column 1 of A078121), then
G(x) = 1 + 2*A(x) + 2^2*A(x)*A(2*x)/2! +
2^3*A(x)*A(2*x)*A(2^2*x)/3! +
2^4*A(x)*A(2*x)*A(2^2*x)*A(2^3*x)/4! + ...

Paul D. Hanna, Table of n, a(n), n== 0..50.

Cf. A078121 (triangle), A002577, A111813 (matrix log); A110505 (q=-1), A111816 (q=3), A111819 (q=4), A111824 (q=5), A111829 (q=6), A111834 (q=7), A111839 (q=8).

{a(n,q=2)=local(A=x/(1-x+x*O(x^n)));for(i=1,n, A=x/(1-x)/(1+sum(j=1,n,prod(k=1,j,subst(A,x,q^k*x))/(j+1)!))); return(n!*polcoeff(A,n))}

E.g.f. satisfies: x/(1-x) = Sum_{n>=1} Prod_{j=0..n-1} A(2^j*x)/(j+1). E.g.f. satisfies: x/(1-x^2) = Sum_{n>=1}Prod_{j=0..2*n}A(2^j*x)/(j+1).

A125794 Column 4 of table A125790; also equals row sums of matrix power A078121^4.

Original entry on oeis.org

1, 5, 25, 165, 1625, 25509, 664665, 29559717, 2290267225, 314039061413, 77160820913241, 34317392762489765, 27859502236825957465, 41575811106337540656037, 114746581654195790543205465, 588765612737696531880325270437, 5642056933026209681424588087899225

Offset: 0

Paul D. Hanna, Dec 10 2006

nonn

Triangle A078121 shifts left one column under matrix square and is related to partitions into powers of 2.

Alois P. Heinz, Table of n, a(n) for n = 0..50

Cf. A125790, A078121; columns: A002577, A125792, A125793, A125795, A125796; diagonals: A125797, A125798.

A diagonal of A152977.

a(n)=local(p=4,q=2,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]))

A111821 Number of partitions of 4*5^n into powers of 5, also equals column 1 of triangle A111820, which shifts columns left and up under matrix 5th power.

Original entry on oeis.org

1, 5, 55, 2055, 291430, 165397680, 390075741430, 3927972221522680, 172358768282285194555, 33479766506261422878944555, 29150234311482124092454001991430

Offset: 0

Gottfried Helms and Paul D. Hanna, Aug 22 2005

nonn

Let q=5; a(n) equals the partitions of (q-1)*q^n into powers of q, or, the coefficient of x^((q-1)*q^n) in 1/Product_{j>=0}(1-x^(q^j)).

Alois P. Heinz, Table of n, a(n) for n = 0..40

Cf. A111820 (triangle), A002577 (q=2), A078124 (q=3), A111817 (q=4), A111826 (q=6), A111831 (q=7), A111836 (q=8).

a(n,q=5)=local(A=Mat(1),B);if(n<0,0, for(m=1,n+2,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(A[n+2,2]))

a(n) = [x^(4*5^n)] 1/Product_{j>=0}(1-x^(5^j)).

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

Formula

A125790 Rectangular table where column k equals row sums of matrix power A078121^k, read by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A125791 a(n) = 2^(n*(n-1)*(n-2)/6) for n>=1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Prolog

Formula

Extensions

A152977 Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) is the number of partitions of 2^n into powers of 2 less than or equal to 2^k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A125792 Column 2 of table A125790; also equals row sums of matrix power A078121^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

A078537 Number of partitions of 4^n into powers of 4 (without regard to order).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A002575 Coefficients of Bell's formula for making change.

Views

A125791 a(n) = 2^(n(n-1)(n-2)/6) for n>=1.