A078537 -id:A078537 - cp's OEIS frontend

A002577 Number of partitions of 2^n into powers of 2.

1, 2, 4, 10, 36, 202, 1828, 27338, 692004, 30251722, 2320518948, 316359580362, 77477180493604, 34394869942983370, 27893897106768940836, 41603705003444309596874, 114788185359199234852802340, 588880400923055731115178072778, 5642645813427132737155703265972004

Offset: 0

N. J. A. Sloane

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 numbers from the second row of T, computed for given m and n > 2, are the (m+2)-gonal numbers. So the second row contains the first members of: A000290 (the square numbers) when m=2, A000326 (the pentagonal numbers) when m=3, and so on. But rows IV, V etc. of the given table are not represented in the OEIS till now. - Valentin Bakoev, Feb 25 2009; edited by M. F. Hasler, Feb 09 2014

			To compute t_2(6,1) we can use a table T, defined as T[i,j]= t_2(i,j), for i=1,2,...,6(=n), and j= 0,1,2,...,32(= k*m^{n-1}). It is: 1,2,3,4,5,6,7,8,9...,33; 1,4,9,16,25,36,49...,81; (so the second row contains the first members of A000290 -- the square numbers) 1,10,35,84,165,...,969; (so the third row contains the first members of A000447. The r-th tetrahedral number is given by formula r(r+1)(r+2)/6. This row (also A000447) contains the tetrahedral numbers, obtained for r=1,3,5,7,...) 1,36,201,656,1625; 1,202,1827; 1,1828; Column 1 contains the first 6 members of A002577. - _Valentin Bakoev_, Feb 25 2009
G.f. = 1 + 2*x + 4*x^2 + 10*x^3 + 36*x^4 + 202*x^5 + 1828*x^6 + ...

R. F. Churchhouse, Binary partitions, pp. 397-400 of A. O. L. Atkin and B. J. Birch, editors, Computers in Number Theory. Academic Press, NY, 1971.
Lawrence, Jim. "Dual-Antiprisms and Partitions of Powers of 2 into Powers of 2." Discrete & Computational Geometry, Vol. 16 (2019): 465-478. See page 466.
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).

Alois P. Heinz, Table of n, a(n) for n = 0..85
V. Bakoev, Algorithmic approach to counting of certain types m-ary partitions, Discrete Mathematics, 275 (2004) pp.17-41.
C. Banderier, H.-K. Hwang, V. Ravelomanana and V. Zacharovas, Analysis of an exhaustive search algorithm in random graphs and the n^{c logn}-asymptotics, 2012. - From _N. J. A. Sloane_, Dec 23 2012
G. Blom and C.-E. Froeberg, Om myntvaexling, (On money-changing) [ Swedish ], Nordisk Matematisk Tidskrift, 10 (1962), 55-69, 103.
G. Blom and C.-E. Froeberg, Om myntvaexling (On money-changing) [Swedish], Nordisk Matematisk Tidskrift, 10 (1962), 55-69, 103. [Annotated scanned copy]
R. F. Churchhouse, Congruence properties of the binary partition function, Math. Proc. Cambr. Phil. Soc. vol 66, no. 2 (1969), 365-370.
Carl-Erik Froberg, Accurate estimation of the number of binary partitions, BIT Numerical Mathematics vol. 17, no 4 (1977) 386-391.
C.-E. Froberg, Accurate estimation of the number of binary partitions [Annotated scanned copy]
H. Minc, The free commutative entropic logarithmetic, Proc. Roy. Soc. Edinburgh Sect. A 65 1959 177-192 (1959).

a(n) = A000123(2^(n-1)) = A018818(2^n).
Cf. A078537, A078121, A000290, A000447.
Column k=2 of A145515, diagonal of A152977. - Alois P. Heinz, Mar 25 2012
See also A002575, A002576.
A column of A125790.
Cf. A000079, A078125, A145513.

import Data.MemoCombinators (memo2, list, integral)
a002577 n = a002577_list !! n
a002577_list = f [1] where
   f xs = (p' xs $ last xs) : f (1 : map (* 2) 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

A002577 := proc(n) if n<=1 then n+1 else A000123(2^(n-1)); fi; end;

$RecursionLimit = 10^5; (* b = A000123 *) b[0] = 1; b[n_?EvenQ] := b[n] = b[n-1] + b[n/2]; b[n_?OddQ] := b[n] = b[n-1] + b[(n-1)/2]; a[n_] := b[2^(n-1)]; a[0] = 1; Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Nov 23 2011 *)
a[ n_] := SeriesCoefficient[ 1 / Product[ 1 - x^2^k, {k, 0, n}], {x, 0, 2^n}]; (* Michael Somos, Apr 21 2014 *)

a(n)=polcoeff(prod(j=0,n,1/(1-x^(2^j)+x*O(x^(2^n)))),2^n) \\ Paul D. Hanna

a(n) is about 0.9233*Sum_j {i=0, 1, 2, 3, ...} 2^(j*(2n-j-1)/2)/j!. - Henry Bottomley, Jul 23 2003
a(n) = A078121(n+1, 1). - Paul D. Hanna, Sep 13 2004
A002577(n)-1 = A125792(n). - Let m > 1, n > 0 and k >= 0. The general formula for the number of all partitions of 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. A002577 is obtained for m=2 and n=1,2,3,... - Valentin Bakoev, Feb 25 2009
a(n) = [x^(2^n)] 1/Product_{j>=0} (1-x^(2^j)). - Alois P. Heinz, Sep 27 2011

Edited by M. F. Hasler, Feb 09 2014

A145515 Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) is the number of partitions of k^n into powers of k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 4, 1, 1, 1, 2, 5, 10, 1, 1, 1, 2, 6, 23, 36, 1, 1, 1, 2, 7, 46, 239, 202, 1, 1, 1, 2, 8, 82, 1086, 5828, 1828, 1, 1, 1, 2, 9, 134, 3707, 79326, 342383, 27338, 1, 1, 1, 2, 10, 205, 10340, 642457, 18583582, 50110484, 692004, 1, 1, 1, 2, 11, 298, 24901, 3649346, 446020582, 14481808030, 18757984046, 30251722, 1, 1

Offset: 0

Alois P. Heinz, Oct 11 2008

			A(2,3) = 5, because there are 5 partitions of 3^2=9 into powers of 3: [1,1,1,1,1,1,1,1,1], [1,1,1,1,1,1,3], [1,1,1,3,3], [3,3,3], [9].
Square array A(n,k) begins:
  1,  1,   1,    1,     1,      1,  ...
  1,  1,   2,    2,     2,      2,  ...
  1,  1,   4,    5,     6,      7,  ...
  1,  1,  10,   23,    46,     82,  ...
  1,  1,  36,  239,  1086,   3707,  ...
  1,  1, 202, 5828, 79326, 642457,  ...

Alois P. Heinz, Antidiagonals n = 0..40, flattened

Columns k=0+1, 2-10 give: A000012, A002577, A078125, A078537, A111822, A111827, A111832, A111837, A145512, A145513.
Row n=3 gives: A189890(k+1).
Main diagonal gives: A145514.
Cf. A007318.

b:= proc(n, j, k) local nn;
      nn:= n+1;
      if n<0  then 0
    elif j=0  or n=0 or k<=1 then 1
    elif j=1  then nn
    elif n>=j then (nn-j) *binomial(nn, j) *add(binomial(j, h)
                   /(nn-j+h) *b(j-h-1, j, k) *(-1)^h, h=0..j-1)
              else b(n, j, k):= b(n-1, j, k) +b(k*n, j-1, k)
      fi
    end:
A:= (n, k)-> b(1, n, k):
seq(seq(A(n, d-n), n=0..d), d=0..13);

b[n_, j_, k_] := Module[{nn = n+1}, Which[n < 0, 0, j == 0 || n == 0 || k <= 1, 1, j == 1, nn, n >= j, (nn-j)*Binomial[nn, j]*Sum[Binomial[j, h]/(nn-j+h)* b[j-h-1, j, k]*(-1)^h, {h, 0, j-1}], True, b[n, j, k] = b[n-1, j, k] + b[k*n, j-1, k] ] ]; a[n_, k_] := b[1, n, k]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 13}] // Flatten (* Jean-François Alcover, Dec 12 2013, translated from Maple *)

See program.

For k>1: A(n,k) = [x^(k^n)] 1/Product_{j>=0} (1-x^(k^j)).

Edited by Alois P. Heinz, Jan 12 2011

A078536 Infinite lower triangular matrix, M, that satisfies [M^4](i,j) = M(i+1,j+1) for all i,j>=0 where [M^n](i,j) denotes the element at row i, column j, of the n-th power of matrix M, with M(0,k)=1 and M(k,k)=1 for all k>=0.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 28, 16, 1, 1, 524, 496, 64, 1, 1, 29804, 41136, 8128, 256, 1, 1, 5423660, 10272816, 2755264, 130816, 1024, 1, 1, 3276048300, 8220685104, 2804672704, 178301696, 2096128, 4096, 1, 1, 6744720496300, 21934062166320, 9139625620672, 729250931456, 11442760704, 33550336, 16384, 1

Offset: 0

Paul D. Hanna, Nov 29 2002

M also satisfies: [M^(4k)](i,j) = [M^k](i+1,j+1) for all i,j,k>=0; thus [M^(4^n)](i,j) = M(i+n,j+n) for all n>=0. Conjecture: sum of the n-th row equals the partitions of 4^n into powers of 4.

			The 4th power of matrix is the same matrix excluding the first row and column:
[1,__0,__0,_0,0]^4=[____1,____0,___0,__0,0]
[1,__1,__0,_0,0]___[____4,____1,___0,__0,0]
[1,__4,__1,_0,0]___[___28,___16,___1,__0,0]
[1,_28,_16,_1,0]___[__524,__496,__64,__1,0]
[1,524,496,64,1]___[29804,41136,8128,256,1]

Cf. A078121, A078122, A078537.

dim = 9;
a[, 0] = 1; a[i, i_] = 1; a[i_, j_] /; j > i = 0;
M = Table[a[i, j], {i, 0, dim-1}, {j, 0, dim-1}];
M4 = MatrixPower[M, 4];
sol = Table[M4[[i, j]] == M[[i+1, j+1]], {i, 1, dim-1}, {j, 1, dim-1}] // Flatten // Solve;
Table[a[i, j], {i, 0, dim-1}, {j, 0, i}] /. sol // Flatten (* Jean-François Alcover, Oct 20 2019 *)

M(n, k) = the coefficient of x^(4^n - 4^(n-k)) in the power series expansion of 1/Product_{j=0..n-k}(1-x^(4^j)) whenever 0<=k0 (conjecture).

More terms from Jean-François Alcover, Oct 20 2019

A111822 Number of partitions of 5^n into powers of 5, also equals the row sums of triangle A111820, which shifts columns left and up under matrix 5th power.

Original entry on oeis.org

1, 2, 7, 82, 3707, 642457, 446020582, 1288155051832, 15905066118254957, 856874264098480364332, 204616369654716156089739332, 219286214391142987407272329973707, 1065403165201779499307991460987124895582, 23663347632778954225192551079067428619449114332

Offset: 0

Gottfried Helms and Paul D. Hanna, Aug 22 2005

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..55

Cf. A111820, A002577 (q=2), A078125 (q=3), A078537 (q=4), A111827 (q=6), A111832 (q=7), A111837 (q=8).

Column k=5 of A145515.

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(sum(k=0,n,A[n+1,k+1])))

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

A111827 Number of partitions of 6^n into powers of 6, also equals the row sums of triangle A111825, which shifts columns left and up under matrix 6th power.

Original entry on oeis.org

1, 2, 8, 134, 10340, 3649346, 6188114528, 52398157106366, 2277627698797283420, 518758596372421679994170, 628925760908337480420110203736, 4109478867142143642923124190955500214

Offset: 0

Gottfried Helms and Paul D. Hanna, Aug 22 2005

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..52

Cf. A111825, A002577 (q=2), A078125 (q=3), A078537 (q=4), A111822 (q=5), A111832 (q=7), A111837 (q=8). Column 6 of A145515.

a(n,q=6)=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(sum(k=0,n,A[n+1,k+1])))

a(n) = [x^(6^n)] 1/Product_{j>=0}(1-x^(6^j)).

A111832 Number of partitions of 7^n into powers of 7, also equals the row sums of triangle A111830, which shifts columns left and up under matrix 7th power.

Original entry on oeis.org

1, 2, 9, 205, 24901, 16077987, 58169810617, 1226373476385199, 154912862345527456431, 119679779055077323244243580, 574461679441277269788798742908435, 17346328772332966415272910459727649244337, 3328366331331467859745524303574824288197338547909

Offset: 0

Gottfried Helms and Paul D. Hanna, Aug 22 2005

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..28

Cf. A111830, A002577 (q=2), A078125 (q=3), A078537 (q=4), A111822 (q=5), A111827 (q=6), A111837 (q=8). Column 7 of A145515.

a(n,q=7)=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(sum(k=0,n,A[n+1,k+1])))

a(n) = [x^(7^n)] 1/Product_{j>=0}(1-x^(7^j)).

A111837 Number of partitions of 8^n into powers of 8, also equals the row sums of triangle A111835, which shifts columns left and up under matrix 8th power.

Original entry on oeis.org

1, 2, 10, 298, 53674, 58573738, 409251498922, 19046062579215274, 6071277235712979102634, 13531779463193107731083553706, 214224474679766323250278564215516074, 24390479071277895100812271376578637910371242, 20173309182842708837666031701435147789403500172143530

Offset: 0

Gottfried Helms and Paul D. Hanna, Aug 22 2005

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..45

Cf. A111835, A002577 (q=2), A078125 (q=3), A078537 (q=4), A111822 (q=5), A111827 (q=6), A111832 (q=7). Column 8 of A145515.

a(n,q=8)=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(sum(k=0,n,A[n+1,k+1])))

a(n) = [x^(8^n)] 1/Product_{j>=0} (1-x^(8^j)).

A111847 Row sums of triangle A111845, which shifts columns left and up under matrix 4th power.

Original entry on oeis.org

1, 2, 9, 97, 2689, 214017, 53130241, 43283609601, 119521939222529, 1144341237628100609, 38638551719263573098497, 4662529388979590206324834305, 2032489532637330252763496597356545

Offset: 0

Paul D. Hanna, Aug 23 2005

nonn

Cf. A111845 (triangle), A078537 (variant).

{a(n,q=4)=local(A=Mat(1),B);if(n<0,0, for(m=1,n+1,B=matrix(m,m);for(i=1,m, for(j=1,i, if(j==i,B[i,j]=1,if(j==1,B[i,j]=(A^q)[i-1,1], B[i,j]=(A^q)[i-1,j-1]));));A=B); return(sum(k=0,n,A[n+1,k+1])))}

A111977 Row sums of triangle A111975, which shifts columns left and up under matrix square.

Original entry on oeis.org

1, 2, 4, 13, 57, 369, 3681, 59073, 1579393, 72188673, 5749089793, 809616264193, 204018868459521, 92907742733348865, 77097057406948106241, 117413997231333438701569, 330195264668839727287861249

Offset: 0

Paul D. Hanna, Aug 25 2005

nonn

Cf. A111975 (triangle), A078537 (variant).

{a(n,q=2)=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,B[i,j]=1,if(j==1,B[i,j]=if(i>2,(A^q)[i-1,2],1), B[i,j]=(A^q)[i-1,j-1]));));A=B); return(sum(k=0,n,A[n+1,k+1])))}

A346565 Number of compositions (ordered partitions) of 4^n into powers of 4.

Original entry on oeis.org

1, 2, 96, 579739960, 773527571233557154337704151068262296

Offset: 0

Ilya Gutkovskiy, Jul 23 2021

nonn

The next term is too large to include.

Index entries for sequences related to compositions

Cf. A000302, A078537, A087221, A248377, A337990, A346564.

Table[SeriesCoefficient[1/(1 - Sum[x^(4^k), {k, 0, n}]), {x, 0, 4^n}], {n, 0, 4}]

a(n) = [x^(4^n)] 1 / (1 - Sum_{k>=0} x^(4^k)).

a(n) = A087221(A000302(n)).

cp's OEIS Frontend

A002577 Number of partitions of 2^n into powers of 2.

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A145515 Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) is the number of partitions of k^n into powers of k.

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A078536 Infinite lower triangular matrix, M, that satisfies [M^4](i,j) = M(i+1,j+1) for all i,j>=0 where [M^n](i,j) denotes the element at row i, column j, of the n-th power of matrix M, with M(0,k)=1 and M(k,k)=1 for all k>=0.

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A111822 Number of partitions of 5^n into powers of 5, also equals the row sums of triangle A111820, which shifts columns left and up under matrix 5th power.

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A111827 Number of partitions of 6^n into powers of 6, also equals the row sums of triangle A111825, which shifts columns left and up under matrix 6th power.

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A111832 Number of partitions of 7^n into powers of 7, also equals the row sums of triangle A111830, which shifts columns left and up under matrix 7th power.

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A111837 Number of partitions of 8^n into powers of 8, also equals the row sums of triangle A111835, which shifts columns left and up under matrix 8th power.

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A111847 Row sums of triangle A111845, which shifts columns left and up under matrix 4th power.

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