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A000009 Expansion of Product_{m=1..infinity} (1 + x^m); number of partitions of n into distinct parts; number of partitions of n into odd parts (if n > 0).
(Formerly M0281 N0100)
387
1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 22, 27, 32, 38, 46, 54, 64, 76, 89, 104, 122, 142, 165, 192, 222, 256, 296, 340, 390, 448, 512, 585, 668, 760, 864, 982, 1113, 1260, 1426, 1610, 1816, 2048, 2304, 2590, 2910, 3264, 3658, 4097, 4582, 5120, 5718, 6378 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Partitions into distinct parts are sometimes called "strict partitions".

The number of different ways to run up a staircase with m steps, taking steps of odd sizes (or taking steps of distinct sizes), where the order is not relevant and there is no other restriction on the number or the size of each step taken is the coefficient of x^m.

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

The result that number of partitions of n into distinct parts = number of partitions of n into odd parts is due to Euler.

Bijection: given n = l1 * 1 + l2 * 3 + l3 * 5 + l7 * 7 + ..., a partition into odd parts, write each li in binary, li = 2^a1 + 2^a2 + 2^a3 + ... where the aj's are all different, then expand n = (2^a1 * 1 + ...)*1 + ... by removing the brackets and we get a partition into distinct parts. For the reverse operation, just keep splitting any even number into halves until no evens remain.

Euler transform of period 2 sequence [1,0,1,0,...]. - Michael Somos, Dec 16, 2002

Number of different partial sums 1+[1,2]+[1,3]+[1,4]+..., where [1,x] indicates a choice. e.g. a(6)=4, as we can write 1+1+1+1+1+1, 1+2+3, 1+2+1+1+1, 1+1+3+1. - Jon Perry, Dec 31 2003

a(n) is the sum of the number of partitions of x_j into at most j parts, where j is the index for the j-th triangular number and n-T(j)=x_j. For example; a(12)=partitions into <=4 parts of 12-T(4)=2 + partitions into <=3 parts of 12-T(3)=6 + partitions into <=2 parts of 12-T(2)=9 + partitions into 1 part of 12-T(1)=11 =(2)(11)+(6)(51)(42)(411)(33)(321)(222)+(9)(81)(72)(63)(54)+(11) =2+7+5+1 =15 - Jon Perry, Jan 13 2004

Number of partitions of n where if k is the largest part, all parts 1..k are present - Jon Perry, Sep 21 2005

a(n) = Sum(A117195(n,k): 0<=k<n) = A117192(n)+A117193(n) for n>0. - Reinhard Zumkeller, Mar 03 2006

The number of connected threshold graphs having n edges. - Michael D. Barrus (mbarrus2(AT)uiuc.edu), Jul 12 2007

Starting with offset 1 = row sums of triangle A146061 and the INVERT transform of A000700 starting: (1, 0, 1, -1, 1, -1, 1, -2, 2, -2, 2, -3, 3, -3, 4, -5,...). - Gary W. Adamson, Oct 26 2008

Number of partitions of n in which the largest part occurs an odd number of times and all other parts occur an even number of times. (Such partitions are the duals of the partitions with odd parts.) - David Wasserman, Mar 04 2009

Equals A035363 convolved with A010054. The convolution square of A000009 = A022567 = A000041 convolved with A010054. A000041 = A000009 convolved with A035363. -  Gary W. Adamson, Jun 11 2009

Considering all partitions of n into distinct parts: there are A140207(n) partitions of maximal size which is A003056(n), and A051162(n) is the greatest number occurring in these partitions. - Reinhard Zumkeller, Jun 13 2009

a(A004526(n)) = A172033(n). - Reinhard Zumkeller, Jan 23 2010

Equals left border of triangle A091602 starting with offset 1. - Gary W. Adamson, Mar 13 2010

a(n) = A054242(n,0) = A201377(n,0). - Reinhard Zumkeller, Dec 02 2011

Number of symmetric unimodal compositions of n+1 where the maximal part appears once. Also number of symmetric unimodal compositions of n where the maximal part appears an odd number of times. - Joerg Arndt, Jun 11 2013

REFERENCES

Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem, Mathematics and Computer Education, Vol. 31, No. 1, pp. 24-28, Winter 1997.

MathEduc Database (Zentralblatt MATH, 1997c.01891).

Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem II, Missouri Journal of Mathematical Sciences, Vol. 16, No. 1, Winter 2004, pp. 12-17. Zentralblatt MATH, Zbl 1071.05501.

G. E. Andrews, The Theory of Partitions, Cambridge University Press, 1998, p. 19.

G. E. Andrews, Number Theory, Dover Publications, 1994, Theorem 12-3, pp. 154-5, and (13-1-1) p. 163.

G. E. Andrews, Euler's "De Partitio Numerorum", Bull. Amer. Math. Soc., 44 (No. 4, 2007), 561-573.

R. Ayoub, An Introduction to the Analytic Theory of Numbers, Amer. Math. Soc., 1963; see p. 196.

T. J. I'a. Bromwich, Introduction to the Theory of Infinite Series, Macmillan, 2nd. ed. 1949, p. 116, Problem 18.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 99.

W. Dunham, The Mathematical Universe, pp. 57-62 J.Wiley 1994.

Leonhard Euler, De partitione numerorum, Novi commentarii academiae scientiarum Petropolitanae 3 (1750/1), 1753, reprinted in: Commentationes Arithmeticae. (Opera Omnia. Series Prima: Opera Mathematica, Volumen Secundum), 1915, Lipsiae et Berolini, 254-294.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 277, Theorems 344, 346.

A. Lascoux, Sylvester's bijection between strict and odd partitions, Discrete Math., 277 (2004), 275-278.

C. J. Moreno and S. S. Wagstaff, Jr., Sums of Squares of Integers, Chapman and Hall, 2006, p. 253.

D. J. Newman, A Problem Seminar, pp. 18;93;102-3 Prob. 93 Springer-Verlag NY 1982.

H. D. Nguyen, D. Taggart, Mining the OEIS: Ten Experimental Conjectures, 2013; http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.391.2522&rep=rep1&type=pdf. Mentions this sequence. - From N. J. A. Sloane, Mar 16 2014

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).

LINKS

N. J. A. Sloane, Table of n, a(n) for n = 0..1999

Joerg Arndt, Fxtbook, pp.348-350

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy], p. 836.

Francesca Aicardi, MATRICIAL FORMULAE FOR PARTITIONS, arXiv:0806.1273.

H. Bottomley, Illustration for A000009, A000041, A047967

Huantian Cao, AutoGF: An Automated System to Calculate Coefficients of Generating Functions.

H. B. C. Darling, Collected Papers of Ramanujan, Table for q(n); n=1 through 100

Alejandro Erickson and Mark Schurch, Monomer-dimer tatami tilings of square regions, Arxiv preprint arXiv:1110.5103, 2011

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 48

E. Georgiadis, Computing Partition Numbers q(n), Technical Report, February (2009). [From Evangelos Georgiadis (egeorg(AT)mit.edu), Mar 19 2009]

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 108

A. Lascoux, Sylvester's bijection

J. Lovejoy, The Number Of Partitions Into Distinct Parts Modulo Powers Of 5

E. Sandifer, How Euler Did It, Philip Naude's problem

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Partition Function P, Partition Function Q, Partition Function bk, Euler Identity, Ramanujan Theta Functions, q-Pochhammer Symbol

Wolfram Research, Generating functions for q(n)

M. P. Zaletel and R. S. K. Mong, Exact Matrix Product States for Quantum Hall Wave Functions, Arxiv preprint arXiv:1208.4862, 2012. - From N. J. A. Sloane, Dec 25 2012

Index entries for "core" sequences

FORMULA

G.f.: prod(m>=1, 1+x^m) = 1/prod(m>=0, 1-x^(2m+1) ) = sum(k>=0, prod(i=1..k, x^i/(1-x^i) ) ) = sum(n>=0, x^(n*(n+1)/2) / prod(k=1..n, 1-x^k) ).

G.f.: sum(n>=0, x^n*prod(k=1,n-1,1+x^k)) = 1+sum(n>=1, x^n*prod(k>=n+1,1+x^k)) - Joerg Arndt, Jan 29 2011

Product_{k=1..inf} (1+x^(2k)) = Sum_{k=0..inf} x^(k*(k+1))/Product_{i=1..k}(1-x^(2i)) - Euler (Hardy and Wright, Theorem 346).

Asymptotics: a(n) ~ exp(pi l_n / sqrt(3)) / ( 4 3^(1/4) l_n^(3/2) ) where l_n = (n-1/24)^(1/2) (Ayoub).

For n>1, a(n)=(1/n)*Sum_{k=1..n} b(k)*a(n-k), with a(0)=1, b(n)= A000593(n)=sum of odd divisors of n; cf. A000700. - Vladeta Jovovic, Jan 21 2002

a(n) = t(n, 0), t as defined in A079211.

a(n)=A026837(n)+A026838(n)=A118301(n)+A118302(n); a(A001318(n))=A051044(n); a(A090864(n))=A118303(n). - Reinhard Zumkeller, Apr 22 2006

Expansion of 1 / chi(-x) = chi(x) / chi(-x^2) = f(-x) / phi(x) = f(x) / phi(-x^2) = psi(x) / f(-x^2) = f(-x^2) / f(-x) = f(-x^4) / psi(-x) in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions. - Michael Somos, Mar 12 2011

G.f. is period 1 Fourier series which satisfies f(-1 / (1152 t)) = 2^(-1/2) / f(t) where q = exp(2 pi i t). - Michael Somos, Aug 16 2007

Expansion of q^(-1/24) * eta(q^2) / eta(q) in powers of q.

Expansion of q^(-1/24) 2^(-1/2) f2(t) in powers of q = exp(2 Pi i t) where f2() is a Weber function. - Michael Somos, Oct 18 2007

Given g.f. A(x), then B(x) = x * A(x^3)^8 satisfies 0 = f(B(x), B(x^2)) where f(u, v) = v - u^2 + 16*u*v^2 . - Michael Somos, May 31 2005

Given g.f. A(x), then B(x) = x * A(x^8)^3 satisfies 0 = f(B(x), B(x^3)) where f(u, v) = (u^3 - v) * (u + v^3) - 9 * u^3 * v^3. - Michael Somos, Mar 25 2008

From Evangelos Georgiadis, Andrew V. Sutherland, Kiran S. Kedlaya (egeorg(AT)mit.edu), Mar 03 2009: (Start)

a(0)=1. a(n)= 2*(Sum_{k=1} (-1)^(k+1) a(n-k^2)) + sigma(n) where

sigma(n)= (-1)^(j) if (n=(j*(3*j+1))/2 OR n=(j*(3*j-1))/2)

otherwise sigma(n)=0. (End)

From Gary W. Adamson, Jun 13 2009: (Start)

The product G.f. = (1/(1-x))*(1/(1-x^3))*(1/(1-x^5))*...; = (1,1,1,...)*

(1,0,0,1,0,0,1,0,0,1,...)*(1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,...) * ...; =

a*b*c*... where a, a*b, a*b*c,...converge to A000009:

1, 1, 1, 2, 2, 2, 3, 3, 3,..4,... = a*b

1, 1, 1, 2, 2, 3, 4, 4, 5,..6,... = a*b*c

1, 1, 1, 2, 2, 3, 4, 5, 6,..7,... = a*b*c*d

1, 1, 1, 2, 2, 3, 4, 5, 6,..8,... = a*b*c*d*e

1, 1, 1, 2, 2, 3, 4, 5, 6,..8,... = a*b*c*d*e*f

...(Cf. analogous example in A000041). (End)

a(n) = P(n)-P(n-2)-P(n-4)+P(n-10)+P(n-14)+...+(-1)^m P(n-2p_m)+..., where P(n) is the partition function (A000041) and p_m =m(3m-1)/2 is the m-th generalized pentagonal number (A001318). - Jerome Malenfant, Feb 16 2011

G.f.: 1/2 (-1; x)_{inf} where (a; q)_k is the q-Pochhammer symbol. [Vladimir Reshetnikov, Apr 24 2013]

EXAMPLE

G.f. = 1 + x + x^2 + 2*x^3 + 2*x^4 + 3*x^5 + 4*x^6 + 5*x^7 + 6*x^8 + 8*x^9 + ...

G.f. = q + q^25 + q^49 + 2*q^73 + 2*q^97 + 3*q^121 + 4*q^145 + 5*q^169 +...

The partitions of n into distinct parts (see A118457) for small n are:

1: 1

2: 2

3: 3, 21

4: 4, 31

5: 5, 41, 32

6: 6, 51, 42, 321

7: 7, 61, 52, 43, 421

8: 8, 71, 62, 53, 521, 431

...

Contribution from Reinhard Zumkeller, Jun 13 2009: (Start)

a(8)=6, A140207(8)=#{5+2+1,4+3+1}=2, A003056(8)=3, A051162(8)=5;

a(9)=8, A140207(9)=#{6+2+1,5+3+1,4+3+2}=3, A003056(9)=3, A051162(9)=6;

a(10)=10, A140207(10)=#{4+3+2+1}=1, A003056(10)=4, A051162(10)=4. (End)

MAPLE

N := 100; t1 := series(mul(1+x^k, k=1..N), x, N); A000009 := proc(n) coeff(t1, x, n); end;

spec := [ P, {P=PowerSet(N), N=Sequence(Z, card>=1)} ]: [ seq(combstruct[count](spec, size=n), n=0..58) ];

spec := [ P, {P=PowerSet(N), N=Sequence(Z, card>=1)} ]: combstruct[allstructs](spec, size=10); # to get the actual partitions for n=10

A := mul(1+x^m, m=1..100); A000009 := n->coeff(A, x, n);

MATHEMATICA

PartitionsQ[Range[0, 60]] (* _Harvey Dale_, Jul 27 2009 *)

a[ n_] := SeriesCoefficient[ Product[1 + x^k, {k, n}], {x, 0, n}]; (* Michael Somos, Jul 06 2011 *)

a[ n_] := SeriesCoefficient[ 1 / Product[1 - x^k, {k, 1, n, 2}], {x, 0, n}]; (* Michael Somos, Jul 06 2011 *)

a[ n_] := With[ {t = Log[q] / (2 Pi I)}, SeriesCoefficient[ q^(-1/24) DedekindEta[2 t] / DedekindEta[ t], {q, 0, n}]]; (* Michael Somos, Jul 06 2011 *)

a[ n_] := SeriesCoefficient[ 1 / QPochhammer[ x, x^2], {x, 0, n}]; (* Michael Somos, May 24 2013 *)

a[ n_] := SeriesCoefficient[ Series[ QHypergeometricPFQ[ {q}, {q x}, q, - q x], {q, 0, n}] /. x -> 1, {q, 0, n}]; (* Michael Somos, Mar 04 2014 *)

a[ n_] := SeriesCoefficient[ QHypergeometricPFQ[{}, {}, q, -1] / 2, {q, 0, n}]; (* Michael Somos, Mar 04 2014 *)

PROG

(PARI) {a(n) = if( n<0, 0, polcoeff( prod( k=1, n, 1 + x^k, 1 + x * O(x^n)), n))}; /* Michael Somos, Nov 17 1999 */

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) / eta(x + A), n))};

(MAGMA) Coefficients(&*[1+x^m:m in [1..100]])[1..100] where x is PolynomialRing(Integers()).1; // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006

(Haskell)

a000009 = p 1 where

   p _ 0 = 1

   p k m = if m < k then 0 else p (k + 1) (m - k) + p (k + 1) m

-- Reinhard Zumkeller, Nov 05 2013

(Maxima) num_distinct_partitions(60, list); /* Emanuele Munarini, Feb 24 2014 */

CROSSREFS

Apart from the first term, equals A052839-1. The rows of A053632 converge to this sequence. When reduced modulo 2 equals the absolute values of A010815. The positions of odd terms given by A001318.

a(n)=sum(A097306(n, m), n=1..m), row sums of triangle of number of partitions of n into m odd parts.

Cf. A000726, A001935, A035959, A219601, A035985, A000041, A000700, A003724, A004111, A007837, A068049, A035294, A078408, A081360, A088670, A109950, A109968, A132312, A146061, A035363, A010054, A091602, A118457 (the partitions), A118459 (partition lengths), A015723 (total number of parts), A230957 (boustrophedon transform).

Sequence in context: A034320 * A081360 A117409 A092833 A100926 A179241

Adjacent sequences:  A000006 A000007 A000008 * A000010 A000011 A000012

KEYWORD

nonn,core,easy,nice

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified April 18 10:47 EDT 2014. Contains 240707 sequences.