A015128 Number of overpartitions of n: an overpartition of n is an ordered sequence of nonincreasing integers that sum to n, where the first occurrence of each integer may be overlined.

1, 2, 4, 8, 14, 24, 40, 64, 100, 154, 232, 344, 504, 728, 1040, 1472, 2062, 2864, 3948, 5400, 7336, 9904, 13288, 17728, 23528, 31066, 40824, 53408, 69568, 90248, 116624, 150144, 192612, 246256, 313808, 398640, 504886, 637592, 802936, 1008448

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

The over-partition function.

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

Also the number of jagged partitions of n.

According to Ramanujan (1913) a(n) is close to (cosh(x)-sinh(x)/x)/(4*n) where x=Pi*sqrt(n). - Michael Somos, Mar 17 2003

Number of partitions of 2n with all odd parts occurring with even multiplicities. There is no restriction on the even parts. Cf. A006950, A046682. - Mamuka Jibladze, Sep 05 2003

Number of partitions of n where there are two kinds of odd parts. - Joerg Arndt, Jul 30 2011. Or, in Gosper's words, partitions into red integers and blue odd integers. - N. J. A. Sloane, Jul 04 2016.

Coincides with the sequence of numbers of nilpotent conjugacy classes in the Lie algebras sp(n), n=0,1,2,3,... (the case n=0 being degenerate). A006950, this sequence and A000041 together cover the nilpotent conjugacy classes in the classical A,B,C,D series of Lie algebras. - Alexander Elashvili, Sep 08 2003

Also, number of 01-partitions of n. A 01-partition of n is a weakly decreasing sequence of m nonnegative integers n(i) such that sum(n(i))=n, n(m)>0, n(j)>=n(j+1)-1 and n(j)>=n(j+2). They are special cases of jagged partitions.

a(8n+7) is divisible by 64 (from Fortin/Jacob/Mathieu paper).

Smallest sequence of even numbers (except a(0)) which is the Euler transform of a sequence of positive integers. - Franklin T. Adams-Watters, Oct 16 2006

Convolution of A000041 and A000009. - Vladeta Jovovic, Nov 26 2002

Equals A022567 convolved with A035363. - Gary W. Adamson, Jun 09 2009

Equals the infinite product [1,2,2,2,...] * [1,0,2,0,2,0,2,...] * [1,0,0,2,0,0,2,0,0,2,...] * ... . - Gary W. Adamson, Jul 05 2009

Equals A182818 convolved with A010815. - Gary W. Adamson, Jul 20 2012

Partial sums of A211971. - Omar E. Pol, Jan 09 2014

Also 1 together with the row sums of A235790. - Omar E. Pol, Jan 19 2014

Antidiagonal sums of A284592. - Peter Bala, Mar 30 2017

The overlining method is equivalent to enumerating the k-subsets of the distinct parts of the i-th partition. - Richard Joseph Boland, Sep 02 2021

Examples

			G.f. = 1 + 2*q + 4*q^2 + 8*q^3 + 14*q^4 + 24*q^5 + 40*q^6 + 64*q^7 + 100*q^8 + ...
For n = 4 the 14 overpartitions of 4 are [4], [4'], [2, 2], [2', 2], [3, 1], [3', 1], [3, 1'], [3', 1'], [2, 1, 1], [2', 1, 1], [2, 1', 1], [2', 1', 1], [1, 1, 1, 1], [1', 1, 1, 1]. - _Omar E. Pol_, Jan 19 2014
		

References

• J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 103.
• R. W. Gosper, Experiments and discoveries in q-trigonometry, in Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics. Editors: F. G. Garvan and M. E. H. Ismail. Kluwer, Dordrecht, Netherlands, 2001, pp. 79-105. See the function g(q).
• James R. Newman, The World of Mathematics, Simon and Schuster, 1956, Vol. I p. 372.

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (first 1000 terms from T. D. Noe)
• Gert Almkvist, Asymptotic formulas and generalized Dedekind sums, Exper. Math., 7 (No. 4, 1998), pp. 343-359.
• Brennan Benfield and Arindam Roy, Log-concavity And The Multiplicative Properties of Restricted Partition Functions, arXiv:2404.03153 [math.NT], 2024.
• Noureddine Chair, Partition identities from Partial Supersymmetry, arXiv:hep-th/0409011, 2004.
• Shi-Chao Chen, On the number of overpartitions into odd parts, Discrete Math. 325 (2014), 32--37. MR3181230.
• William Y.C. Chen and Ernest X.W. Xia, Proof of a conjecture of Hirschhorn and Sellers on overpartitions, arXiv:1307.4155 [math.CO], 2013; Acta Arith. 163 (2014), no. 1, 59--69. MR3194057
• Sylvie Corteel, Particle seas and basic hypergeometric series, Advances Appl. Math., 31 (2003), 199-214.
• Sylvie Corteel and Jeremy Lovejoy, Frobenius partitions and the combinatorics of Ramanujan's 1 psi 1 summation, J. Combin. Theory A 97 (2002), 177-183.
• Sylvie Corteel and Jeremy Lovejoy, Overpartitions, Trans. Amer. Math. Soc., 356 (2004), 1623-1635.
• Brian Drake, Limits of areas under lattice paths, Discrete Math. 309 (2009), no. 12, 3936-3953.
• Benjamin Engel, Log-concavity of the overpartition function, The Ramanujan Journal, Vol. 43, No. 2 (2017), pp. 229-241; arXiv preprint, arXiv:1412.4603 [math.NT], 2014.
• Alex Fink, Richard K. Guy and Mark Krusemeyer, Partitions with parts occurring at most thrice, Contributions to Discrete Mathematics, Vol 3, No 2 (2008).
• J.-F. Fortin, P. Jacob and P. Mathieu, Jagged partitions, The Ramanujan Journal, Vol. 10, No. 2 (2005), pp. 215-235; arXiv preprint, arXiv:math/0310079 [math.CO], 2003-2005.
• Frank Garvan, Table of a(n) for n = 1..10000.
• R. W. Gosper, Experiments and discoveries in q-trigonometry, in F. G. Garvan and M. E. H. Ismail (eds.), Symbolic computation, number theory, special functions, physics and combinatorics, Springer, Boston, MA, 2001, pp. 79-105; preprint.
• R. W. Gosper, q-Trigonometry: Some Prefatory Afterthoughts
• William J. Keith, Restricted k-color partitions, The Ramanujan Journal, Vol. 40, No. 1 (2016), pp. 71-92; arXiv preprint, arXiv:1408.4089 [math.CO], 2014.
• Byungchan Kim, A short note on the overpartition function, Discr. Math., 309 (2009), 2528-2532.
• Byungchan Kim, Overpartition pairs modulo powers of 2, Discrete Math., 311 (2011), 835-840.
• Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016.
• Jeremy Lovejoy, Gordon's theorem for overpartitions, J. Combin. Theory A 103 (2003), 393-401.
• Karl Mahlburg, The overpartition function modulo small powers of 2, Discr. Math., 286 (2004), 263-267.
• Mircea Merca, Overpartitions in terms of 2-adic valuation, Aequat. Math. (2024). See p. 9.
• Igor Pak, Partition bijections, a survey, The Ramanujan Journal, Vol. 12, No. 1 (2006), pp. 5-75; alternative link.
• Maxie D. Schmidt, Exact Formulas for the Generalized Sum-of-Divisors Functions, arXiv:1705.03488 [math.NT], 2017-2019. See Example 4.1, p. 13.
• Michael Somos, Introduction to Ramanujan theta functions
• Liuquan Wang, Another Proof of a Conjecture by Hirschhorn and Sellers on Overpartitions, J. Int. Seq. 17 (2014) # 14.9.8.
• Eric Weisstein's World of Mathematics, Modular Equation.
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
• Michael P. Zaletel and Roger S. K. Mong, Exact matrix product states for quantum Hall wave functions, Physical Review B, Vol. 86, No. 24 (2012), 245305; arXiv preprint, arXiv:1208.4862 [cond-mat.str-el], 2012. - From _N. J. A. Sloane_, Dec 25 2012

Crossrefs

Cf. A022567, A035363, A002448, A106507, A156616, A261386, A265835, A014968, A284592, A303662.

See A004402 for a version with signs.

Column k=2 of A321884.

Cf. A002513.

Programs

Julia

# JacobiTheta4 is defined in A002448.
A015128List(len) = JacobiTheta4(len, -1)
A015128List(40) |> println # Peter Luschny, Mar 12 2018

Maple

mul((1+x^n)/(1-x^n),n=1..256): seq(coeff(series(%,x,n+1),x,n), n=0..40);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1) +2*add(b(n-i*j, i-1), j=1..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..40);  # Alois P. Heinz, Feb 10 2014
a_list := proc(len) series(1/JacobiTheta4(0,x),x,len+1); seq(coeff(%,x,j),j=0..len) end: a_list(39); # Peter Luschny, Mar 14 2017

Mathematica

max = 39; f[x_] := Exp[Sum[(DivisorSigma[1, 2*n] - DivisorSigma[1, n])*(x^n/n), {n, 1, max}]]; CoefficientList[ Series[f[x], {x, 0, max}], x] (* Jean-François Alcover, Jun 11 2012, after Joerg Arndt *)
a[ n_] := SeriesCoefficient[ QHypergeometricPFQ[ {-1}, {}, x, x], {x, 0, n}]; (* Michael Somos, Mar 11 2014 *)
QP = QPochhammer; s = QP[q^2]/QP[q]^2 + O[q]^40; CoefficientList[s + O[q]^100, q] (* Jean-François Alcover, Nov 25 2015, after Michael Somos *)
Table[Sum[PartitionsP[n-k]*PartitionsQ[k], {k, 0, n}], {n, 0, 50}] (* Vaclav Kotesovec, Nov 28 2015 *)
(QPochhammer[-x, x]/QPochhammer[x, x] + O[x]^50)[[3]] (* Vladimir Reshetnikov, Nov 12 2016 *)
nmax = 100; p = ConstantArray[0, nmax+1]; p[[1]] = 1; Do[p[[n+1]] = 0; k = 1; While[n + 1 - k^2 > 0, p[[n+1]] += (-1)^(k+1)*p[[n + 1 - k^2]]; k++;]; p[[n+1]] = 2*p[[n+1]];, {n, 1, nmax}]; p (* Vaclav Kotesovec, Apr 11 2017 *)
a[ n_] := SeriesCoefficient[ 1 / EllipticTheta[ 4, 0, x], {x, 0, n}]; (* Michael Somos, Nov 15 2018 *)
a[n_] := Sum[2^Length[Union[IntegerPartitions[n][[i]]]], {i, 1, PartitionsP[n]}]; (* Richard Joseph Boland, Sep 02 2021 *)
n = 39; CoefficientList[Product[(1 + x^k)/(1 - x^k), {k, 1, n}] + O[x]^(n + 1), x] (* Oliver Seipel, Sep 19 2021 *)

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) / eta(x + A)^2, n))}; /* Michael Somos, Nov 01 2008 */

PARI

{a(n)=polcoeff(exp(sum(m=1,n\2+1,2*x^(2*m-1)/(1-x^(2*m-1)+x*O(x^n))/(2*m-1))),n)} /* Paul D. Hanna, Aug 06 2009 */

PARI

N=66; x='x+O('x^N); gf=exp(sum(n=1,N,(sigma(2*n)-sigma(n))*x^n/n));Vec(gf) /* Joerg Arndt, Jul 30 2011 */

PARI

lista(nn) = {q='q+O('q^nn); Vec(eta(q^2)/eta(q)^2)} \\ Altug Alkan, Mar 20 2018

SageMath

# uses[EulerTransform from A166861]
a = BinaryRecurrenceSequence(0, 1, 1, 2)
b = EulerTransform(a)
print([b(n) for n in range(40)]) # Peter Luschny, Nov 11 2020

Formula

Euler transform of period 2 sequence [2, 1, ...]. - Michael Somos, Mar 17 2003

G.f.: Product_{m>=1} (1 + q^m)/(1 - q^m).

G.f.: 1 / (Sum_{m=-inf..inf} (-q)^(m^2)) = 1/theta_4(q).

G.f.: 1 / Product_{m>=1} (1 - q^(2*m)) * (1 - q^(2*m-1))^2.

G.f.: exp( Sum_{n>=1} 2*x^(2*n-1)/(1 - x^(2*n-1))/(2*n-1) ). - Paul D. Hanna, Aug 06 2009

G.f.: exp( Sum_{n>=1} (sigma(2*n) - sigma(n))*x^n/n ). - Joerg Arndt, Jul 30 2011

G.f.: Product_{n>=0} theta_3(q^(2^n))^(2^n). - Joerg Arndt, Aug 03 2011

A004402(n) = (-1)^n * a(n). - Michael Somos, Mar 17 2003

Expansion of eta(q^2) / eta(q)^2 in powers of q. - Michael Somos, Nov 01 2008

Expansion of 1 / phi(-q) in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Nov 01 2008

Convolution inverse of A002448. - Michael Somos, Nov 01 2008

Recurrence: a(n) = 2*Sum_{m>=1} (-1)^(m+1) * a(n-m^2).

a(n) = (1/n)*Sum_{k=1..n} (sigma(2*k) - sigma(k))*a(n-k). - Vladeta Jovovic, Dec 05 2004

G.f.: Product_{i>=1} (1 + x^i)^A001511(2i) (see A000041). - Jon Perry, Jun 06 2004

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = w^4 * (u^4 + v^4) - 2 * u^2 * v^6. - Michael Somos, Nov 01 2008

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u6^3 * (u1^2 + u3^2) - 2 * u1 * u2 * u3^3. - Michael Somos, Nov 01 2008

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u2^3 * (u3^2 - 3 * u1^2) + 2 * u1^3 * u3 * u6. - Michael Somos, Nov 01 2008

G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 32^(-1/2) (t/i)^(-1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A106507. - Michael Somos, Nov 01 2008

a(n) = 2*A014968(n), n >= 1. - Omar E. Pol, Jan 19 2014

a(n) ~ Pi * BesselI(3/2, Pi*sqrt(n)) / (4*sqrt(2)*n^(3/4)). - Vaclav Kotesovec, Jan 11 2017

Let T(n,k) = the number of partitions of n with parts 1 through k of two kinds, T(n,0) = A000041(n), the number of partitions of n. Then a(n) = T(n,0) + T(n-1,1) + T(n-3,2) + T(n-6,3) + T(n-10,4) + T(n-15,5) + ... . Gregory L. Simay, May 29 2019

For n >= 1, a(n) = Sum_{k>=1} 2^k * A116608(n,k). - Gregory L. Simay, Jun 01 2019

Sum_{n>=1} 1/a(n) = A303662. - Amiram Eldar, Nov 15 2020

a(n) = Sum_{i=1..p(n)} 2^(d(n,i)), where d(n,i) is the number of distinct parts in the i-th partition of n. - Richard Joseph Boland, Sep 02 2021

G.f.: A(x) = exp( Sum_{n >= 1} x^n*(2 + x^n)/(n*(1 - x^(2*n))) ). - Peter Bala, Dec 23 2021

G.f. A(q) satisfies (3*A(q)/A(q^9) - 1)^3 = 9*A(q)^4/A(q^3)^4 - 1. - Paul D. Hanna, Oct 14 2024

Extensions

Minor edits by Vaclav Kotesovec, Sep 13 2014