A018819 - OEIS

OFFSET

0,3

COMMENTS

First differences of A000123; also A000123 with terms repeated. See the relevant proof that follows the first formula below.

Among these partitions there is exactly one partition with all distinct terms, as every number can be expressed as the sum of the distinct powers of 2.

Euler transform of A036987 with offset 1.

a(n) is the number of "non-squashing" partitions of n, that is, partitions n = 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. - N. J. A. Sloane, Nov 30 2003

Normally the OEIS does not include sequences like this where every term is repeated, but an exception was made for this one because of its importance. The unrepeated sequence A000123 is the main entry.

Number of different partial sums from 1 + [1, *2] + [1, *2] + ..., where [1, *2] means we can either add 1 or multiply by 2. E.g., a(6) = 6 because we have 6 = 1 + 1 + 1 + 1 + 1 + 1 = (1+1) * 2 + 1 + 1 = 1 * 2 * 2 + 1 + 1 = (1+1+1) * 2 = 1 * 2 + 1 + 1 + 1 + 1 = (1*2+1) * 2 where the connection is defined via expanding each bracket; e.g., this is 6 = 1 + 1 + 1 + 1 + 1 + 1 = 2 + 2 + 1 + 1 = 4 + 1 + 1 = 2 + 2 + 2 = 2 + 1 + 1 + 1 + 1 = 4 + 2. - Jon Perry, Jan 01 2004

Number of partitions p of n such that the number of compositions generated by p is odd. For proof see the Alekseyev and Adams-Watters link. - Vladeta Jovovic, Aug 06 2007

Differs from A008645 first at a(64). - R. J. Mathar, May 28 2008

Appears to be row sums of A155077. - Mats Granvik, Jan 19 2009

Number of partitions (p_1, p_2, ..., p_k) of n, with p_1 >= p_2 >= ... >= p_k, such that for each i, p_i >= p_{i+1} + ... + p_k. - John MCKAY (mckay(AT)encs.concordia.ca), Mar 06 2009 (these are the "non-squashing" partitions as nonincreasing lists).

Equals rightmost diagonal of triangle of A168261. Starting with offset 1 = eigensequence of triangle A115361 and row sums of triangle A168261. - Gary W. Adamson, Nov 21 2009

Equals convolution square root of A171238: (1, 2, 5, 8, 16, 24, 40, 56, 88, ...). - Gary W. Adamson, Dec 05 2009

Let B = the n-th convolution power of the sequence and C = the aerated variant of B. It appears that B/C = the binomial sequence beginning (1, n, ...). Example: Third convolution power of the sequence is (1, 3, 9, 19, 42, 78, 146, ...), with C = (1, 0, 3, 0, 9, 0, 19, ...). Then B/C = (1, 3, 6, 10, 15, 21, ...). - Gary W. Adamson, Aug 15 2016

From Gary W. Adamson, Sep 08 2016: (Start)

The limit of the matrix power M^k as n-->inf results in a single column vector equal to the sequence, where M is the following production matrix:

1, 0, 0, 0, 0, ...

1, 1, 0, 0, 0, ...

1, 1, 1, 0, 0, ...

1, 1, 1, 1, 0, ...

1, 1, 1, 1, 1, ...

... (End)

a(n) is the number of "non-borrowing" partitions of n, meaning binary subtraction of a smaller part from a larger part will never require place-value borrowing. - David V. Feldman, Jan 29 2020

From Gus Wiseman, May 25 2024: (Start)

Also the number of multisets of positive integers whose binary rank is n, where the binary rank of a multiset m is given by Sum_i 2^(m_i-1). For example, the a(1) = 1 through a(8) = 10 multisets are:

{1} {2} {12} {3} {13} {23} {123} {4}

{11} {111} {22} {122} {113} {1113} {33}

{112} {1112} {222} {1222} {223}

{1111} {11111} {1122} {11122} {1123}

{11112} {111112} {2222}

{111111} {1111111} {11113}

{11222}

{111122}

{1111112}

{11111111}

(End)

LINKS

T. D. Noe, Table of n, a(n) for n = 0..1000

Max Alekseyev and Franklin T. Adams-Watters, Two proofs of an observation of Vladeta Jovovic

Giedrius Alkauskas, Generalization of the Rodseth-Gupta theorem on binary partitions, Lithuanian Math. J., 43 (2) (2003), 103-110.

Giedrius Alkauskas, Congruence Properties of the Function that Counts Compositions into Powers of 2 , J. Int. Seq. 13 (2010), 10.5.3.

Joerg Arndt, Matters Computational (The Fxtbook), section 38.1, p.729.

Scott M. Bailey and Donald M. Larson, The A(1)-module structure of the homology of Brown-Gitler spectra, arXiv:2107.01316 [math.AT], 2021.

Valentin P. Bakoev, Algorithmic approach to counting of certain types m-ary partitions, Discrete Mathematics, 275 (2004) pp. 17-41.

Philippe Biane, Laver tables and combinatorics, arXiv:1810.00548 [math.CO], 2018. Mentions this sequence.

Peter J. Cameron, Firdous Ee Jannat, Rajat Kanti Nath, and Reza Sharafdini, A survey on conjugacy class graphs of groups, arXiv:2403.09423 [math.GR], 2024.

Karl Dilcher and Larry Ericksen, Polynomial Analogues of Restricted b-ary Partition Functions, J. Int. Seq., Vol. 22 (2019), Article 19.3.2.

Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, 2009; see page 48, 581.

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.

Michael D. Hirschhorn and James A. Sellers, A different view of m-ary partitions, Australasian J. Combin., vol.30 (2004), 193-196.

Jonathan Jordan and Richard Southwell, Further Properties of Reproducing Graphs, Applied Mathematics, Vol. 1 No. 5, 2010, pp. 344-350. doi: 10.4236/am.2010.15045. - From N. J. A. Sloane, Feb 03 2013

Yasuyuki Kachi and Pavlos Tzermias, On the m-ary partition numbers, Algebra and Discrete Mathematics, Volume 19 (2015). Number 1, pp. 67-76.

Matjaž Konvalinka and Igor Pak, Cayley compositions, partitions, polytopes, and geometric bijections, Journal of Combinatorial Theory, Series A, Volume 123, Issue 1, April 2014, Pages 86-91; see also DOI link. - From N. J. A. Sloane, Dec 22 2012

Apisit Pakapongpun and Thomas Ward, Functorial Orbit counting, J. Int. Seq., 12 (2009) 09.2.4, example 25.

Øystein J. Rodseth and James A. Sellers, On a Restricted m-Non-Squashing Partition Function, Journal of Integer Sequences, Vol. 8 (2005), Article 05.5.4.

David Ruelle, Dynamical zeta functions and transfer operators, Notices Amer. Math. Soc., 49 (No. 8, 2002), 887-895; see p. 888.

N. J. A. Sloane and James A. Sellers, On non-squashing partitions, arXiv:math/0312418 [math.CO], 2003.

N. J. A. Sloane and James A. Sellers, On non-squashing partitions, Discrete Math., 294 (2005), 259-274.

FORMULA

a(2m+1) = a(2m), a(2m) = a(2m-1) + a(m). Proof: If n is odd there is a part of size 1; removing it gives a partition of n - 1. If n is even either there is a part of size 1, whose removal gives a partition of n - 1, or else all parts have even sizes and dividing each part by 2 gives a partition of n/2.

G.f.: 1 / Product_{j>=0} (1-x^(2^j)).

a(n) = (1/n)*Sum_{k = 1..n} A038712(k)*a(n-k), n > 1, a(0) = 1. - Vladeta Jovovic, Aug 22 2002

a(2*n) = a(2*n + 1) = A000123(n). - Michael Somos, Aug 25 2003

a(n) = 1 if n = 0, Sum_{j = 0..floor(n/2)} a(j) if n > 0. - David W. Wilson, Aug 16 2007

G.f. A(x) satisfies A(x^2) = (1-x) * A(x). - Michael Somos, Aug 25 2003

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^2*w - 2*u*v^2 + v^3. - Michael Somos, Apr 10 2005

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u6 * u1^3 - 3*u3*u2*u1^2 + 3*u3*u2^2*u1 - u3*u2^3. - Michael Somos, Oct 15 2006

G.f.: 1/( Sum_{n >= 0} x^evil(n) - x^odious(n) ), where evil(n) = A001969(n) and odious(n) = A000069(n). - Paul D. Hanna, Jan 23 2012

Let A(x) by the g.f. and B(x) = A(x^k), then 0 = B*((1-A)^k - (-A)^k) + (-A)^k, see fxtbook link. - Joerg Arndt, Dec 17 2012

G.f.: Product_{n>=0} (1+x^(2^n))^(n+1), see the fxtbook link. - Joerg Arndt, Feb 28 2014

G.f.: 1 + Sum_{i>=0} x^(2^i) / Product_{j=0..i} (1 - x^(2^j)). - Ilya Gutkovskiy, May 07 2017

EXAMPLE

G.f. = 1 + x + 2*x^2 + 2*x^3 + 4*x^4 + 4*x^5 + 6*x^6 + 6*x^7 + 10*x^8 + ...

a(4) = 4: the partitions are 4, 2 + 2, 2 + 1 + 1, 1 + 1 + 1 + 1.

a(7) = 6: the partitions are 4 + 2 + 1, 4 + 1 + 1 + 1, 2 + 2 + 2 + 1, 2 + 2 + 1 + 1 + 1, 2 + 1 + 1 + 1 + 1 + 1, 1 + 1 + 1 + 1 + 1 + 1 + 1.

From Joerg Arndt, Dec 17 2012: (Start)

The a(10) = 14 binary partitions of 10 are (in lexicographic order)

[ 1] [ 1 1 1 1 1 1 1 1 1 1 ]

[ 2] [ 2 1 1 1 1 1 1 1 1 ]

[ 3] [ 2 2 1 1 1 1 1 1 ]

[ 4] [ 2 2 2 1 1 1 1 ]

[ 5] [ 2 2 2 2 1 1 ]

[ 6] [ 2 2 2 2 2 ]

[ 7] [ 4 1 1 1 1 1 1 ]

[ 8] [ 4 2 1 1 1 1 ]

[ 9] [ 4 2 2 1 1 ]

[10] [ 4 2 2 2 ]

[11] [ 4 4 1 1 ]

[12] [ 4 4 2 ]

[13] [ 8 1 1 ]

[14] [ 8 2 ]

The a(11) = 14 binary partitions of 11 are obtained by appending 1 to each partition in the list.

The a(10) = 14 non-squashing partitions of 10 are (in lexicographic order)

[ 1] [ 6 3 1 1 ]

[ 2] [ 6 3 2 ]

[ 3] [ 6 4 1 ]

[ 4] [ 6 5 ]

[ 5] [ 7 2 1 1 ]

[ 6] [ 7 2 2 ]

[ 7] [ 7 3 1 ]

[ 8] [ 7 4 ]

[ 9] [ 8 2 1 ]

[10] [ 8 3 ]

[11] [ 9 1 1 ]

[12] [ 9 2 ]

[13] [ 10 1 ]

[14] [ 11 ]

The a(11) = 14 non-squashing partitions of 11 are obtained by adding 1 to the first part in each partition in the list.

(End)

From David V. Feldman, Jan 29 2020: (Start)

The a(10) = 14 non-borrowing partitions of 10 are (in lexicographic order)

[ 1] [1 1 1 1 1 1 1 1 1 1]

[ 2] [2 2 2 2 2]

[ 3] [3 1 1 1 1 1 1 1]

[ 4] [3 3 1 1 1 1]

[ 5] [3 3 2 2]

[ 6] [3 3 3 1]

[ 7] [5 1 1 1 1 1]

[ 8] [5 5]

[ 9] [6 2 2]

[10] [6 4]

[11] [7 1 1 1]

[12] [7 3]

[13] [9 1]

[14] [10]

The a(11) = 14 non-borrowing partitions of 11 are obtained either by adding 1 to the first even part in each partition (if any) or else appending a 1 after the last part.

(End)

For example, the five partitions of 4, written in nonincreasing order, are [1, 1, 1, 1], [2, 1, 1], [2, 2], [3, 1], [4]. The last four satisfy the condition, and a(4) = 4. The Maple program below verifies this for small values of n.

MAPLE

with(combinat); N:=8; a:=array(1..N); c:=array(1..N);

for n from 1 to N do p:=partition(n); np:=nops(p); t:=0;

for s to np do r:=p[s]; r:=sort(r, `>`); nr:=nops(r); j:=1;

# while j<nr and r[j]>sum(r[k], k=j+1..nr) do j:=j+1; od; # gives A040039

while j<nr and r[j]>= sum(r[k], k=j+1..nr) do j:=j+1; od; # gives A018819

if j=nr then t:=t+1; fi od; a[n]:=t; od; # John McKay

MATHEMATICA

max = 59; a[0] = a[1] = 1; a[n_?OddQ] := a[n] = a[n-1]; a[n_?EvenQ] := a[n] = a[n-1] + a[n/2]; Table[a[n], {n, 0, max}]

(* or *) CoefficientList[Series[1/Product[(1-x^(2^j)), {j, 0, Log[2, max] // Ceiling}], {x, 0, max}], x] (* Jean-François Alcover, May 17 2011, updated Feb 17 2014 *)

a[ n_] := If[n<1, Boole[n==0], a[n] = a[n-1] + If[EvenQ@n, a[Quotient[n, 2]], 0]]; (* Michael Somos, May 04 2022 *)

Table[Count[IntegerPartitions[n], _?(AllTrue[Log2[#], IntegerQ]&)], {n, 0, 60}] (* Harvey P. Dale, Jun 20 2024 *)

PROG

(PARI) { n=15; v=vector(n); for (i=1, n, v[i]=vector(2^(i-1))); v[1][1]=1; for (i=2, n, k=length(v[i-1]); for (j=1, k, v[i][j]=v[i-1][j]+1; v[i][j+k]=v[i-1][j]*2)); c=vector(n); for (i=1, n, for (j=1, 2^(i-1), if (v[i][j]<=n, c[v[i][j]]++))); c } /* Jon Perry */

(PARI) {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)); polcoeff(A, n))}; /* Michael Somos, Aug 25 2003 */

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

(Haskell)

a018819 n = a018819_list !! n

a018819_list = 1 : f (tail a008619_list) where

f (x:xs) = (sum $ take x a018819_list) : f xs

-- Reinhard Zumkeller, Jan 28 2012

(Haskell)

import Data.List (intersperse)

a018819 = (a018819_list !!)

a018819_list = 1 : 1 : (<*>) (zipWith (+)) (intersperse 0) (tail a018819_list)

-- Johan Wiltink, Nov 08 2018

(Python)