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%I #252 Apr 05 2024 08:20:39
%S 1,2,4,8,14,24,40,64,100,154,232,344,504,728,1040,1472,2062,2864,3948,
%T 5400,7336,9904,13288,17728,23528,31066,40824,53408,69568,90248,
%U 116624,150144,192612,246256,313808,398640,504886,637592,802936,1008448
%N 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.
%C The over-partition function.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C Also the number of jagged partitions of n.
%C 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
%C 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
%C 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.
%C 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
%C 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.
%C a(8n+7) is divisible by 64 (from Fortin/Jacob/Mathieu paper).
%C 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
%C Convolution of A000041 and A000009. - _Vladeta Jovovic_, Nov 26 2002
%C Equals A022567 convolved with A035363. - _Gary W. Adamson_, Jun 09 2009
%C 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
%C Equals A182818 convolved with A010815. - _Gary W. Adamson_, Jul 20 2012
%C Partial sums of A211971. - _Omar E. Pol_, Jan 09 2014
%C Also 1 together with the row sums of A235790. - _Omar E. Pol_, Jan 19 2014
%C Antidiagonal sums of A284592. - _Peter Bala_, Mar 30 2017
%C 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
%D J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 103.
%D 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).
%D James R. Newman, The World of Mathematics, Simon and Schuster, 1956, Vol. I p. 372.
%H Vaclav Kotesovec, <a href="/A015128/b015128.txt">Table of n, a(n) for n = 0..10000</a> (first 1000 terms from T. D. Noe)
%H Gert Almkvist, <a href="https://projecteuclid.org/euclid.em/1047674152">Asymptotic formulas and generalized Dedekind sums</a>, Exper. Math., 7 (No. 4, 1998), pp. 343-359.
%H Brennan Benfield and Arindam Roy, <a href="https://arxiv.org/abs/2404.03153">Log-concavity And The Multiplicative Properties of Restricted Partition Functions</a>, arXiv:2404.03153 [math.NT], 2024.
%H Noureddine Chair, <a href="http://arXiv.org/abs/hep-th/0409011">Partition identities from Partial Supersymmetry</a>, arXiv:hep-th/0409011, 2004.
%H Shi-Chao Chen, <a href="http://dx.doi.org/10.1016/j.disc.2014.02.015">On the number of overpartitions into odd parts</a>, Discrete Math. 325 (2014), 32--37. MR3181230.
%H William Y.C. Chen and Ernest X.W. Xia, <a href="http://arxiv.org/abs/1307.4155">Proof of a conjecture of Hirschhorn and Sellers on overpartitions</a>, arXiv:1307.4155 [math.CO], 2013; Acta Arith. 163 (2014), no. 1, 59--69. MR3194057
%H Sylvie Corteel, <a href="http://dx.doi.org/10.1016/S0196-8858(03)00011-3">Particle seas and basic hypergeometric series</a>, Advances Appl. Math., 31 (2003), 199-214.
%H Sylvie Corteel and Jeremy Lovejoy, <a href="http://dx.doi.org/10.1006/jcta.2001.3205">Frobenius partitions and the combinatorics of Ramanujan's 1 psi 1 summation</a>, J. Combin. Theory A 97 (2002), 177-183.
%H Sylvie Corteel and Jeremy Lovejoy, <a href="http://dx.doi.org/10.1090/S0002-9947-03-03328-2">Overpartitions</a>, Trans. Amer. Math. Soc., 356 (2004), 1623-1635.
%H Brian Drake, <a href="http://dx.doi.org/10.1016/j.disc.2008.11.020">Limits of areas under lattice paths</a>, Discrete Math. 309 (2009), no. 12, 3936-3953.
%H Benjamin Engel, <a href="https://doi.org/10.1007/s11139-015-9762-0">Log-concavity of the overpartition function</a>, The Ramanujan Journal, Vol. 43, No. 2 (2017), pp. 229-241; <a href="https://arxiv.org/abs/1412.4603">arXiv preprint</a>, arXiv:1412.4603 [math.NT], 2014.
%H Alex Fink, Richard K. Guy and Mark Krusemeyer, <a href="https://doi.org/10.11575/cdm.v3i2.61940">Partitions with parts occurring at most thrice</a>, Contributions to Discrete Mathematics, Vol 3, No 2 (2008).
%H J.-F. Fortin, P. Jacob and P. Mathieu, <a href="https://doi.org/10.1007/s11139-005-4848-8">Jagged partitions</a>, The Ramanujan Journal, Vol. 10, No. 2 (2005), pp. 215-235; <a href="http://arxiv.org/abs/math.CO/0310079">arXiv preprint</a>, arXiv:math/0310079 [math.CO], 2003-2005.
%H Frank Garvan, <a href="https://qseries.org/fgarvan/data/optncofs.txt">Table of a(n) for n = 1..10000</a>.
%H R. W. Gosper, <a href="https://doi.org/10.1007/978-1-4613-0257-5_6">Experiments and discoveries in q-trigonometry</a>, 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; <a href="/A274621/a274621.pdf">preprint</a>.
%H R. W. Gosper, <a href="/A274621/a274621_1.pdf">q-Trigonometry: Some Prefatory Afterthoughts</a>
%H William J. Keith, <a href="https://doi.org/10.1007/s11139-015-9704-x">Restricted k-color partitions</a>, The Ramanujan Journal, Vol. 40, No. 1 (2016), pp. 71-92; <a href="https://arxiv.org/abs/1408.4089">arXiv preprint</a>, arXiv:1408.4089 [math.CO], 2014.
%H Byungchan Kim, <a href="http://dx.doi.org/10.1016/j.disc.2008.05.007">A short note on the overpartition function</a>, Discr. Math., 309 (2009), 2528-2532.
%H Byungchan Kim, <a href="http://dx.doi.org/10.1016/j.disc.2011.02.002">Overpartition pairs modulo powers of 2</a>, Discrete Math., 311 (2011), 835-840.
%H Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], 2015-2016.
%H Jeremy Lovejoy, <a href="http://dx.doi.org/10.1016/S0097-3165(03)00116-X">Gordon's theorem for overpartitions</a>, J. Combin. Theory A 103 (2003), 393-401.
%H Karl Mahlburg, <a href="http://dx.doi.org/10.1016/j.disc.2004.03.014">The overpartition function modulo small powers of 2</a>, Discr. Math., 286 (2004), 263-267.
%H Igor Pak, <a href="https://doi.org/10.1007/s11139-006-9576-1">Partition bijections, a survey</a>, The Ramanujan Journal, Vol. 12, No. 1 (2006), pp. 5-75; <a href="https://www.math.ucla.edu/~pak/papers/psurvey.pdf">alternative link</a>.
%H Maxie D. Schmidt, <a href="https://arxiv.org/abs/1705.03488">Exact Formulas for the Generalized Sum-of-Divisors Functions</a>, arXiv:1705.03488 [math.NT], 2017-2019. See Example 4.1, p. 13.
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Liuquan Wang, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Wang2/wang15.html">Another Proof of a Conjecture by Hirschhorn and Sellers on Overpartitions</a>, J. Int. Seq. 17 (2014) # 14.9.8.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%H Michael P. Zaletel and Roger S. K. Mong, <a href="https://doi.org/10.1103/PhysRevB.86.245305">Exact matrix product states for quantum Hall wave functions</a>, Physical Review B, Vol. 86, No. 24 (2012), 245305; <a href="http://arxiv.org/abs/1208.4862">arXiv preprint</a>, arXiv:1208.4862 [cond-mat.str-el], 2012. - From _N. J. A. Sloane_, Dec 25 2012
%F Euler transform of period 2 sequence [2, 1, ...]. - _Michael Somos_, Mar 17 2003
%F G.f.: Product_{m>=1} (1 + q^m)/(1 - q^m).
%F G.f.: 1 / (Sum_{m=-inf..inf} (-q)^(m^2)) = 1/theta_4(q).
%F G.f.: 1 / Product_{m>=1} (1 - q^(2*m)) * (1 - q^(2*m-1))^2.
%F 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
%F G.f.: exp( Sum_{n>=1} (sigma(2*n) - sigma(n))*x^n/n ). - _Joerg Arndt_, Jul 30 2011
%F G.f.: Product_{n>=0} theta_3(q^(2^n))^(2^n). - _Joerg Arndt_, Aug 03 2011
%F A004402(n) = (-1)^n * a(n). - _Michael Somos_, Mar 17 2003
%F Expansion of eta(q^2) / eta(q)^2 in powers of q. - _Michael Somos_, Nov 01 2008
%F Expansion of 1 / phi(-q) in powers of q where phi() is a Ramanujan theta function. - _Michael Somos_, Nov 01 2008
%F Convolution inverse of A002448. - _Michael Somos_, Nov 01 2008
%F Recurrence: a(n) = 2*Sum_{m>=1} (-1)^(m+1) * a(n-m^2).
%F a(n) = (1/n)*Sum_{k=1..n} (sigma(2*k) - sigma(k))*a(n-k). - _Vladeta Jovovic_, Dec 05 2004
%F G.f.: Product_{i>=1} (1 + x^i)^A001511(2i) (see A000041). - _Jon Perry_, Jun 06 2004
%F 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
%F 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
%F 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
%F 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
%F a(n) = 2*A014968(n), n >= 1. - _Omar E. Pol_, Jan 19 2014
%F a(n) ~ Pi * BesselI(3/2, Pi*sqrt(n)) / (4*sqrt(2)*n^(3/4)). - _Vaclav Kotesovec_, Jan 11 2017
%F 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
%F For n >= 1, a(n) = Sum_{k>=1} 2^k * A116608(n,k). - _Gregory L. Simay_, Jun 01 2019
%F Sum_{n>=1} 1/a(n) = A303662. - _Amiram Eldar_, Nov 15 2020
%F 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
%F G.f.: A(x) = exp( Sum_{n >= 1} x^n*(2 + x^n)/(n*(1 - x^(2*n))) ). - _Peter Bala_, Dec 23 2021
%e 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 + ...
%e 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
%p mul((1+x^n)/(1-x^n),n=1..256): seq(coeff(series(%,x,n+1),x,n), n=0..40);
%p # second Maple program:
%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p b(n, i-1) +2*add(b(n-i*j, i-1), j=1..n/i)))
%p end:
%p a:= n-> b(n$2):
%p seq(a(n), n=0..40); # _Alois P. Heinz_, Feb 10 2014
%p 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
%t 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_ *)
%t a[ n_] := SeriesCoefficient[ QHypergeometricPFQ[ {-1}, {}, x, x], {x, 0, n}]; (* _Michael Somos_, Mar 11 2014 *)
%t 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_ *)
%t Table[Sum[PartitionsP[n-k]*PartitionsQ[k], {k, 0, n}], {n, 0, 50}] (* _Vaclav Kotesovec_, Nov 28 2015 *)
%t (QPochhammer[-x, x]/QPochhammer[x, x] + O[x]^50)[[3]] (* _Vladimir Reshetnikov_, Nov 12 2016 *)
%t 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 *)
%t a[ n_] := SeriesCoefficient[ 1 / EllipticTheta[ 4, 0, x], {x, 0, n}]; (* _Michael Somos_, Nov 15 2018 *)
%t a[n_] := Sum[2^Length[Union[IntegerPartitions[n][[i]]]], {i, 1, PartitionsP[n]}]; (* _Richard Joseph Boland_, Sep 02 2021 *)
%t n = 39; CoefficientList[Product[(1 + x^k)/(1 - x^k), {k, 1, n}] + O[x]^(n + 1), x] (* _Oliver Seipel_, Sep 19 2021 *)
%o (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 */
%o (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 */
%o (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 */
%o (PARI) lista(nn) = {q='q+O('q^nn); Vec(eta(q^2)/eta(q)^2)} \\ _Altug Alkan_, Mar 20 2018
%o (Julia) # JacobiTheta4 is defined in A002448.
%o A015128List(len) = JacobiTheta4(len, -1)
%o A015128List(40) |> println # _Peter Luschny_, Mar 12 2018
%o (SageMath) # uses[EulerTransform from A166861]
%o a = BinaryRecurrenceSequence(0, 1, 1, 2)
%o b = EulerTransform(a)
%o print([b(n) for n in range(40)]) # _Peter Luschny_, Nov 11 2020
%Y Cf. A022567, A035363, A002448, A106507, A156616, A261386, A265835, A014968, A284592, A303662.
%Y See A004402 for a version with signs.
%Y Column k=2 of A321884.
%Y Cf. A002513.
%K nonn,easy,nice
%O 0,2
%A _N. J. A. Sloane_
%E Minor edits by _Vaclav Kotesovec_, Sep 13 2014
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