A089177 -id:A089177 - cp's OEIS frontend

A000123 Number of binary partitions: number of partitions of 2n into powers of 2.

1, 2, 4, 6, 10, 14, 20, 26, 36, 46, 60, 74, 94, 114, 140, 166, 202, 238, 284, 330, 390, 450, 524, 598, 692, 786, 900, 1014, 1154, 1294, 1460, 1626, 1828, 2030, 2268, 2506, 2790, 3074, 3404, 3734, 4124, 4514, 4964, 5414, 5938, 6462, 7060, 7658, 8350, 9042, 9828

Offset: 0

N. J. A. Sloane

Also, a(n) = number of "non-squashing" partitions of 2n (or 2n+1), that is, partitions 2n = p_1 + p_2 + ... + p_k with 1 <= p_1 <= p_2 <= ... <= p_k and p_1 + p_2 + ... + p_i <= p_{i+1} for all 1 <= i < k [Hirschhorn and Sellers].
Row sums of A101566. - Paul Barry, Jan 03 2005
Equals infinite convolution product of [1,2,2,2,2,2,2,2,2] aerated A000079 - 1 times, i.e., [1,2,2,2,2,2,2,2,2] * [1,0,2,0,2,0,2,0,2] * [1,0,0,0,2,0,0,0,2]. - Mats Granvik and Gary W. Adamson, Aug 04 2009
Which can be further decomposed to the infinite convolution product of finally supported sequences, namely [1,1] aerated A000079 - 1 times with multiplicity A000027 + 1 times, i.e., [1,1] * [1,1] * [1,0,1] * [1,0,1] * [1,0,1] * ... (next terms are [1,0,0,0,1] 4 times, etc.). - Eitan Y. Levine, Jun 18 2023
Given A018819 = A000123 with repeats, polcoeff (1, 1, 2, 2, 4, 4, ...) * (1, 1, 1, ...) = (1, 2, 4, 6, 10, ...) = (1, 0, 2, 0, 4, 0, 6, ...) * (1, 2, 2, 2, ...). - Gary W. Adamson, Dec 16 2009
Let M = an infinite lower triangular matrix with (1, 2, 2, 2, ...) in every column shifted down twice. A000123 = lim_{n->infinity} M^n, the left-shifted vector considered as a sequence. Replacing (1, 2, 2, 2, ...) with (1, 3, 3, 3, ...) and following the same procedure, we obtain A171370: (1, 3, 6, 12, 18, 30, 42, 66, 84, 120, ...). - Gary W. Adamson, Dec 06 2009
First differences of the sequence are (1, 2, 2, 4, 4, 6, 6, 10, ...), A018819, i.e., the sequence itself with each term duplicated except for the first one (unless a 0 is prefixed before taking the first differences), as shown by the formula a(n) - a(n-1) = a(floor(n/2)), valid for all n including n = 0 if we let a(-1) = 0. - M. F. Hasler, Feb 19 2019
Sum over k <= n of number of partitions of k into powers of 2, A018819. - Peter Munn, Feb 21 2020

			For non-squashing partitions and binary partitions see the example in A018819.
For n=3, the a(3)=6 admitted partitions of 2n=6 are 1+1+1+1+1+1, 1+1+1+1+2, 1+1+2+2, 2+2+2, 1+1+4 and 2+4. - _R. J. Mathar_, Aug 11 2021

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976.
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.
N. G. de Bruijn, On Mahler's partition problem, Indagationes Mathematicae, vol. X (1948), 210-220.
G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
H. Gupta, A simple proof of the Churchhouse conjecture concerning binary partitions, Indian J. Pure Appl. Math. 3 (1972), 791-794.
H. Gupta, A direct proof of the Churchhouse conjecture concerning binary partitions, Indian J. Math. 18 (1976), 1-5.
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..65536 (first 10001 terms from T. D. Noe)
Joerg Arndt, Matters Computational (The Fxtbook), p. 728
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
Sara Billey, Matjaž Konvalinka and Frederick A. Matsen IV, On trees, tanglegrams, and tangled chains, hal-02173394 [math.CO], 2020.
Henry Bottomley, Illustration of initial terms
N. G. de Bruijn, On Mahler's partition problem, 1948.
R. F. Churchhouse, Congruence properties of the binary partition function, Proc. Cambridge Philos. Soc. 66 1969 371-376.
Philippe Deléham, Letter to N. J. A. Sloane, Apr 20 1998
P. Dumas and P. Flajolet, Asymptotique des recurrences mahleriennes: le cas cyclotomique, Journal de Théorie des Nombres de Bordeaux 8 (1996), pp. 1-30.
Amanda Folsom et al, On a general class of non-squashing partitions, Discrete Mathematics 339.5 (2016): 1482-1506.
C.-E. Froberg, Accurate estimation of the number of binary partitions, Nordisk Tidskr. Informationsbehandling (BIT) 17 (1977), 386-391.
C.-E. Froberg, Accurate estimation of the number of binary partitions [Annotated scanned copy]
Maciej Gawron, Piotr Miska and Maciej Ulas, Arithmetic properties of coefficients of power series expansion of Prod_{n>=0} (1-x^(2^n))^t, arXiv:1703.01955 [math.NT], 2017.
H. Gupta, Proof of the Churchhouse conjecture concerning binary partitions, Proc. Camb. Phil. Soc. 70 (1971), 53-56.
R. K. Guy, Letters to N. J. A. Sloane and J. W. Moon, 1988
M. D. Hirschhorn and J. A. Sellers, A different view of m-ary partitions, Australasian J. Combin., 30 (2004), 193-196.
M. D. Hirschhorn and J. A. Sellers, A different view of m-ary partitions
Youkow Homma, Jun Hwan Ryu and Benjamin Tong, Sequence non-squashing partitions, Slides from a talk, Jul 24 2014.
K. Ji and H. S. Wilf, Extreme palindromes, Amer. Math. Monthly, 115, no. 5 (2008), 447-451.
Y. Kachi and P. Tzermias, On the m-ary partition numbers, Algebra and Discrete Mathematics, Volume 19 (2015). Number 1, pp. 67-76.
Martin Klazar, What is an answer? — remarks, results and problems on PIO formulas in combinatorial enumeration, part I, arXiv:1808.08449 [math.CO], 2018.
D. E. Knuth, An almost linear recurrence, Fib. Quart., 4 (1966), 117-128.
M. Konvalinka and I. Pak, Cayley compositions, partitions, polytopes, and geometric bijections - _N. J. A. Sloane_, Dec 22 2012
M. Konvalinka and I. Pak, Cayley compositions, partitions, polytopes, and geometric bijections, Journal of Combinatorial Theory, Series A, Volume 123, Issue 1, April 2014, Pages 86-91.
Vaclav Kotesovec, Graph - the asymptotic ratio (10^8 terms)
M. Latapy, Partitions of an integer into powers, DMTCS Proceedings AA (DM-CCG), 2001, 215-228.
M. Latapy, Partitions of an integer into powers, DMTCS Proceedings AA (DM-CCG), 2001, 215-228. [Cached copy, with permission]
K. Mahler, On a special functional equation, Journ. London Math. Soc. 15 (1940), 115-123.
Mathematics Stack Exchange, Efficient computation of number of partitions into powers of 2, Jul 10 2024.
E. O'Shea, M-partitions: optimal partitions of weight for one scale pan, Discrete Math. 289 (2004), 81-93. See Lemma 29.
Igor Pak, Complexity problems in enumerative combinatorics, arXiv:1803.06636 [math.CO], 2018.
John L. Pfaltz, Evaluating the binary partition function when N = 2^n, Congr. Numer, 109:3-12, 1995. [Broken link]
B. Reznick, Some binary partition functions, in "Analytic number theory" (Conf. in honor P. T. Bateman, Allerton Park, IL, 1989), 451-477, Progr. Math., 85, Birkhäuser Boston, Boston, MA, 1990.
O. J. Rodseth, Enumeration of M-partitions, Discrete Math., 306 (2006), 694-698.
O. J. Rodseth and J. A. Sellers, Binary partitions revisited, J. Combinatorial Theory, Series A 98 (2002), 33-45.
O. J. Rodseth and J. A. Sellers, On a Restricted m-Non-Squashing Partition Function, Journal of Integer Sequences, Vol. 8 (2005), Article 05.5.4.
D. Ruelle, Dynamical zeta functions and transfer operators, Notices Amer. Math. Soc., 49 (No. 8, 2002), 887-895; see p. 888.
Frank Ruskey, Info on binary partitions
N. J. A. Sloane and J. A. Sellers, On non-squashing partitions, arXiv:math/0312418 [math.CO], 2003.
N. J. A. Sloane and J. A. Sellers, On non-squashing partitions, Discrete Math., 294 (2005), 259-274.
Daniel G. Zhu, An improved lower bound on the Shannon capacities of complements of odd cycles, arXiv:2402.10025 [math.CO], 2024.
Index entries for "core" sequences

Cf. A000041, A002033, A002487, A002577, A005704-A005706, A023359, A040039, A100529. Partial sums and bisection of A018819.
A column of A072170. Row sums of A089177. Twice A033485.
Cf. A145515. - Alois P. Heinz, Apr 16 2009
Cf. A171370. - Gary W. Adamson, Dec 06 2009

import Data.List (transpose)
a000123 n = a000123_list !! n
a000123_list = 1 : zipWith (+)
   a000123_list (tail $ concat $ transpose [a000123_list, a000123_list])
-- Reinhard Zumkeller, Nov 15 2012, Aug 01 2011

[1] cat [n eq 1 select n+1 else Self(n-1) + Self(n div 2): n in [1..70]]; // Vincenzo Librandi, Dec 17 2016

A000123 := proc(n) option remember; if n=0 then 1 else A000123(n-1)+A000123(floor(n/2)); fi; end; [ seq(A000123(i),i=0..50) ];
# second Maple program: more efficient for large n; try: a( 10^25 );
g:= proc(b, n) option remember; `if`(b<0, 0, `if`(b=0 or
      n=0, 1, `if`(b>=n, add((-1)^(t+1)*binomial(n+1, t)
      *g(b-t, n), t=1..n+1), g(b-1, n)+g(2*b, n-1))))
    end:
a:= n-> (t-> g(n/2^(t-1), t))(max(ilog2(2*n), 1)):
seq(a(n), n=0..60); # Alois P. Heinz, Apr 16 2009, revised Apr 14 2016

a[0] = 1; a[n_] := a[n] = a[Floor[n/2]] + a[n-1]; Array[a,49,0] (* Jean-François Alcover, Apr 11 2011, after M. F. Hasler *)
Fold[Append[#1, Total[Take[Flatten[Transpose[{#1, #1}]], #2]]] &, {1}, Range[2, 49]] (* Birkas Gyorgy, Apr 18 2011 *)

{a(n) = my(A, m); if( n<1, n==0, m=1; A = 1 + O(x); while(m<=n, m*=2; A = subst(A, x, x^2) * (1+x) / (1-x)); polcoeff(A, n))}; /* Michael Somos, Aug 25 2003 */

{a(n) = if( n<1, n==0, a(n\2) + a(n-1))}; /* Michael Somos, Aug 25 2003 */

A123=[];A000123(n)={ n<3 && return(2^n); if( n<=#A123, A123[n] && return(A123[n]); A123[n-1] && return( A123[n] = A123[n-1]+A000123(n\2) ), n>2*#A123 && A123=concat(A123,vector((n-#A123)\2))); A123[if(n>#A123,1,n)]=2*sum(k=1,n\2-1,A000123(k),1)+(n%2+1)*A000123(n\2)} \\ Stores results in global vector A123 dynamically resized to at most 3n/4 when size is less than n/2. Gives a(n*10^6) in ~ n sec. - M. F. Hasler, Apr 30 2009

{a(n)=polcoeff(exp(sum(m=1,n,2^valuation(2*m,2)*x^m/m)+x*O(x^n)),n)} \\ Paul D. Hanna, Oct 30 2012

from functools import lru_cache
@lru_cache(maxsize=None)
def A000123(n): return 1 if n == 0 else A000123(n-1) + A000123(n//2) # Chai Wah Wu, Jan 18 2022

a(n) = A018819(2*n).
a(n) = a(n-1) + a(floor(n/2)). For proof see A018819.
2 * a(n) = a(n+1) + a(n-1) if n is even. - Michael Somos, Jan 07 2011
G.f.: (1-x)^(-1) Product_{n>=0} (1 - x^(2^n))^(-1).
a(n) = Sum_{i=0..n} a(floor(i/2)) [O'Shea].
a(n) = (1/n)*Sum_{k=1..n} (A038712(k)+1)*a(n-k), n > 1, a(0)=1. - Vladeta Jovovic, Aug 22 2002
Conjecture: Limit_{n ->infinity} (log(n)*a(2n))/(n*a(n)) = c = 1.63... - Benoit Cloitre, Jan 26 2003 [The constant c is equal to 2*log(2) = 1.38629436... =A016627. - Vaclav Kotesovec, Aug 07 2019]
G.f. A(x) satisfies A(x^2) = ((1-x)/(1+x)) * A(x). - Michael Somos, Aug 25 2003
G.f.: Product_{k>=0} (1+x^(2^k))/(1-x^(2^k)) = (Product_{k>=0} (1+x^(2^k))^(k+1) )/(1-x) = Product_{k>=0} (1+x^(2^k))^(k+2). - Joerg Arndt, Apr 24 2005
From Philippe Flajolet, Sep 06 2008: (Start)
The asymptotic rate of growth is known precisely - see De Bruijn's paper. With p(n) the number of partitions of n into powers of two, the asymptotic formula of de Bruijn is: log(p(2*n)) = 1/(2*L2)*(log(n/log(n)))^2 + (1/2 + 1/L2 + LL2/L2)*log(n) - (1 + LL2/L2)*log(log(n)) + Phi(log(n/log(n))/L2), where L2=log(2), LL2=log(log(2)) and Phi(x) is a certain periodic function with period 1 and a tiny amplitude.
Numerically, Phi(x) appears to have a mean value around 0.66. An expansion up to O(1) term had been obtained earlier by Kurt Mahler. (End)
G.f.: exp( Sum_{n>=1} 2^A001511(n) * x^n/n ), where 2^A001511(n) is the highest power of 2 that divides 2*n. - Paul D. Hanna, Oct 30 2012
(n/2)*a(n) = Sum_{k = 0..n-1} (n-k)/A000265(n-k)*a(k). - Peter Bala, Mar 03 2019
Conjectures from Mikhail Kurkov, May 04 2025: (Start)
Sum_{k=0..n} a(2^m*k)*A106400(n-k) = A125790(m,2*n) for m >= 0, n >= 0.
Sum_{k=0..n} a(2^m*(2*k+1))*A106400(n-k) = A125790(m+1,2*n+1) for m >= 0, n >= 0.
More generally, if we define b(n,m,p,q) = Sum_{k=0..n} a(2^m*(2*p*k+2*q+1))*A106400(n-k) for m >= 0, p > 0, q >= 0, n >= 0, then it also looks like that we have b(n,m,p,q) = Sum_{k=0..m+1} A078121(m+1,k)*b(n,k,p/2,(q-1)/2), b(n,m,p,q) = Sum_{k=0..m+1} A078121(m+1,k)*b(n,k,p/2,q/2)*(-1)^(m+k+1) for m >= 0, p > 0, q >= 0, n >= 0. (End)
Conjecture: Sum_{i>=0} a(2^m*i + k)*x^i = f(k,x) / Product_{q>=0} (1 - x^(2^q)) for m > 0, 2^(m-1) <= k < 2^m where f(k,x) is g.f. for k-th row of A381810. - Mikhail Kurkov, May 17 2025

More terms from Robin Trew (trew(AT)hcs.harvard.edu)

Values up to a(10^4) checked with given PARI code by M. F. Hasler, Apr 30 2009

A078121 Infinite lower triangular matrix, M, that satisfies [M^2](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, 2, 1, 1, 4, 4, 1, 1, 10, 16, 8, 1, 1, 36, 84, 64, 16, 1, 1, 202, 656, 680, 256, 32, 1, 1, 1828, 8148, 10816, 5456, 1024, 64, 1, 1, 27338, 167568, 274856, 174336, 43680, 4096, 128, 1, 1, 692004, 5866452, 11622976, 8909648, 2794496, 349504, 16384, 256, 1

Offset: 0

Paul D. Hanna, Nov 18 2002

M also satisfies: [M^(2k)](i,j) = [M^k](i+1,j+1) for all i,j,k>=0; thus [M^(2^n)](i,j) = M(i+n,j+n) for all n>=0.

			The square of the matrix is the same matrix excluding the first row and column:
  [1, 0, 0, 0, 0]^2 = [ 1, 0, 0, 0, 0]
  [1, 1, 0, 0, 0]     [ 2, 1, 0, 0, 0]
  [1, 2, 1, 0, 0]     [ 4, 4, 1, 0, 0]
  [1, 4, 4, 1, 0]     [10,16, 8, 1, 0]
  [1,10,16, 8, 1]     [36,84,64,16, 1]

Alois P. Heinz, Rows n = 0..80, flattened
MathOverflow, Closed form for diagonals of A078121, (2025).

Cf. A002577, A016131, A078122, A089177.

M:= proc(i, j) option remember; `if`(j=0 or i=j, 1,
       add(M(i-1, k)*M(k, j-1), k=0..i-1))
    end:
seq(seq(M(n,k), k=0..n), n=0..10);  # Alois P. Heinz, Feb 27 2015

rows = 10; M[k_] := Table[ Which[j == 1, 1, i == j, 1, 1 < j < i, m[i, j], True, 0], {i, 1, k}, {j, 1, k}]; m2[i_, j_] := m[i+1, j+1]; M2[k_] := Table[ Which[jJean-François Alcover, Feb 27 2015 *)
M[i_, j_] := M[i, j] = If[j == 0 || i == j, 1, Sum[M[i-1, k]*M[k, j-1], {k, 0, i-1}]]; Table[Table[M[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Feb 27 2015, after Alois P. Heinz *)

rows_upto(n) = my(A, v1); v1 = vector(n+1, i, vector(i, j, 0)); v1[1][1] = 1; for(i=1, n, v1[i+1][1] = 1; v1[i+1][i+1] = 1); for(i=2, n, for(j=1, i-1, A = (i+j+1)%2; v1[i+1][j+1] = 2*sum(k=0, (i-j-1)\2, v1[i-j+1][2*k+A+1]*v1[j+2*k+A+1][j]))); v1 \\ Mikhail Kurkov, Aug 27 2025

M(1,j) = A002577(j) (partitions of 2^j into powers of 2), M(j+1,j) = 2^j, M(j+2,j) = 4^j, M(j+3,j) = A016131(j).
M(n,k) = the coefficient of x^(2^n - 2^(n-k)) in the power series expansion of 1/Product_{j=0..n-k} (1-x^(2^j)) whenever 0<=k0 (conjecture).
M(n,k) = Sum_{j=0..n-k-1} M(n-k,j)*M(k+j,k-1)*(1+(-1)^(n+k+j+1)) for 0 < k < n with M(n,0) = M(n,n) = 1. - Mikhail Kurkov, Jun 01 2025
From Mikhail Kurkov, Jul 01 2025: (Start)
Conjecture 1: let R(n,x) be the n-th row polynomial, then R(n,x) = x*R(n-1,x) + Sum_{k=1..n-1} M(n-1,k-1)*R(k,x)*(-1)^(n+k+1) = R(n-1,x) + x*Sum_{k=1..n-1} (M(n-1,k) - M(n-2,k))*R(k,x) for n > 1 with R(0,x) = 1, R(1,x) = x + 1.
Conjecture 2: M(n+m,n) ~ 2^(m*(2*n+m-1)/2)/m! as n -> oo. More generally, it also looks like that M(n+m,n) for m > 0 can be represented as (Sum_{j=0..flooor((m-1)/2)} 2^((m-2*j)*(2*(n-j)+m-1)/2)*P(m,j)*(-1)^j)/m! where P(m,j) are some positive integer coefficients. (End)

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A000123 Number of binary partitions: number of partitions of 2n into powers of 2.

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A078121 Infinite lower triangular matrix, M, that satisfies [M^2](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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A181322 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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A089178 Triangle T(n,k) (n >= 0, 0 <= k <= 1+log_2(floor(n+1))) read by rows: row 0 = {1}, row 1 = {1 1}; for n >=2, row n = row n-1 + (row floor((n-1)/2) shifted one place right).

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