A125792 -id:A125792 - 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

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

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.

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

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]))

A125793 Column 3 of table A125790; also equals row sums of matrix power A078121^3.

Original entry on oeis.org

1, 4, 16, 84, 656, 8148, 167568, 5866452, 356855440, 38315189204, 7352635371152, 2547660633170900, 1607532367023451792, 1860491404939092059092, 3974085151281967171382928, 15751822048486986712162264020

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.

Cf. A125790, A078121; columns: A002577, A125792, A125794, A125795, A125796; diagonals: A125797, A125798.

a(n)=local(p=3,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]))

A125795 Column 5 of table A125790; also equals row sums of matrix power A078121^5.

Original entry on oeis.org

1, 6, 36, 286, 3396, 64350, 2026564, 109082974, 10243585092, 1704787839326, 509106367263812, 275575947307878750, 272638898948894782532, 496470192421055920965982, 1674003944602430578138969156, 10505662319550964196499807897950, 123269344114733507237294056110191684

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.

Cf. A125790, A078121; columns: A002577, A125792, A125793, A125794, A125796; diagonals: A125797, A125798.

a(n)=local(p=5,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]))

A125796 Column 6 of table A125790; also equals row sums of matrix power A078121^6.

Original entry on oeis.org

1, 7, 49, 455, 6321, 140231, 5174449, 326603719, 35994670257, 7036275790791, 2470183452677297, 1573137497080468423, 1832597507832323118257, 3932481446278522861786055, 15637033863127787477309461681, 115814953429924513361085880079303, 1604893891765170672173387008222303409

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.

Cf. A125790, A078121; columns: A002577, A125792, A125793, A125794, A125795; diagonals: A125797, A125798.

a(n)=local(p=6,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]))

A125797 Main diagonal of table A125790.

Original entry on oeis.org

1, 2, 9, 84, 1625, 64350, 5174449, 841185704, 275723872209, 181906966455026, 241258554545388985, 642662865556736504700, 3436011253857466940820073, 36852501476559726217536067974, 792571351187806816558255494473185

Offset: 0

Paul D. Hanna, Dec 10 2006

nonn

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.

Cf. A125790, A078121; columns: A002577, A125792, A125793, A125794, A125795, A125796; A125798 (diagonal).

a(n)=local(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^n)[n+1,c+1]))

A125798 A diagonal of table A125790: a(n) = A125790(n+1,n).

Original entry on oeis.org

1, 4, 35, 656, 25509, 2026564, 326603719, 106355219008, 69808185542089, 92203545302072964, 244779396712068825067, 1305009502037405316440848, 13963029918525356899170492525, 299675759834305402824238609624548

Offset: 0

Paul D. Hanna, Dec 10 2006

nonn

Cf. A125790, A078121; columns: A002577, A125792, A125793, A125794, A125795, A125796; A125797 (diagonal).

a(n)=local(q=2,A=Mat(1), B); 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(c=0,n+1,(A^n)[n+2,c+1]))

cp's OEIS Frontend

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

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

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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.

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A002575 Coefficients of Bell's formula for making change.

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

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A125793 Column 3 of table A125790; also equals row sums of matrix power A078121^3.

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A125795 Column 5 of table A125790; also equals row sums of matrix power A078121^5.

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A125796 Column 6 of table A125790; also equals row sums of matrix power A078121^6.

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