A022567 Expansion of Product_{m>=1} (1+x^m)^2.

1, 2, 3, 6, 9, 14, 22, 32, 46, 66, 93, 128, 176, 238, 319, 426, 562, 736, 960, 1242, 1598, 2048, 2608, 3306, 4175, 5248, 6570, 8198, 10190, 12622, 15589, 19190, 23552, 28830, 35190, 42842, 52034, 63040, 76198, 91904, 110604, 132832, 159216, 190464, 227417

Offset: 0

Author

N. J. A. Sloane, Jun 14 1998

Keywords

• nonn

Comments

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

Number of partitions of n into distinct parts, with 2 types of each part. E.g., for n=4, we consider k and k* to be different versions of k and so we have 4, 4*, 31, 31*, 3*1, 3*1*, 22*, 211*, 2*11*, thus a(4)=9. - Jon Perry, Apr 04 2004

Number of partitions of n into odd parts, each part being of two kinds. E.g., a(3)=6 because we have 3, 3', 1+1+1, 1+1+1', 1+1'+1', 1'+1'+1'. - Emeric Deutsch, Mar 22 2005

Euler transform of period 2 sequence [2,0,2,0,...]. - Emeric Deutsch, Mar 22 2005

Equals A000041 convolved with A010054. - Gary W. Adamson, Jun 11 2009

The sum of the least gaps in all partitions of n. The "least gap" of a partition is the least positive integer that is not a part of the partition. Example: a(4) = 9 because the least gaps in [4], [3,1], [2,2], [2,1,1], and [1,1,1,1] are 1, 2, 1, 3, and 2, respectively. - Emeric Deutsch, May 18 2015

Number of 2-regular bipartitions of n. - N. J. A. Sloane, Oct 20 2019

The least gap is also known as the minimal excludant or mex; see Andrews and Newman. - George Beck, Dec 10 2020

Examples

			G.f. = 1 + 2*x + 3*x^2 + 6*x^3 + 9*x^4 + 14*x^5 + 22*x^6 + 32*x^7 + 46*x^8 + ...
G.f. = q + 2*q^13 + 3*q^25 + 6*q^37 + 9*q^49 + 14*q^61 + 22*q^73 + 32*q^85 + ...
		

References

• P. J. Grabner, A. Knopfmacher, Analysis of some new partition statistics, Ramanujan J., 12, 2006, 439-454.
• Kathiravan, T., and S. N. Fathima. "On L-regular bipartitions modulo L." The Ramanujan Journal 44.3 (2017): 549-558.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000
• George E. Andrews, David Newman, Partitions and the Minimal Excludant, Annals of Combinatorics, Volume 23, May 2019, Pages 249-254.
• Cristina Ballantine, Mircea Merca, Bisected theta series, least r-gaps in partitions, and polygonal numbers, arXiv:1710.05960 [math.CO], 2017.
• Helena Bergold, Lukas Egeling, and Hung. P. Hoang, Signotopes with few plus signs, arXiv:2411.19208 [math.CO], 2024. See p. 14.
• J. Currie, N. Rampersad, Binary words avoiding xx^Rx and strongly unimodal sequences, JIS 18 (2015) #15.10.3.
• Alejandro Erickson, Frank Ruskey, Enumerating maximal tatami mat coverings of square grids with v vertical dominoes, arXiv:1304.0070 [math.CO], 2013.
• Alejandro Erickson and Mark Schurch, Monomer-dimer tatami tilings of square regions, arXiv preprint arXiv:1110.5103 [math.CO], 2011.
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 852
• Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 8.
• Mircea Merca, A new look on the generating function for the number of divisors, Journal of Number Theory, Volume 149, April 2015, Pages 57-69.
• Mircea Merca, Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer, Journal of Number Theory, Volume 160, March 2016, Pages 60-75. See q(n)'.
• Mbavhalelo Mulokwe and Konstantinos Zoubos, Free fermions, neutrality and modular transformations, arXiv:2403.08531 [hep-th], 2024.
• Michael Somos, Introduction to Ramanujan theta functions
• Jacob Sprittulla, On Colored Factorizations, arXiv:2008.09984 [math.CO], 2020.
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A000009, A022597, A285221, A304048.

Cf. A010054. - Gary W. Adamson, Jun 11 2009

Column k=2 of A286335.

Number of r-regular bipartitions of n for r = 2,3,4,5,6: A022567, A328547, A001936, A263002, A328548.

Programs

Magma

Coefficients(&*[(1+x^m)^2:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 26 2018

Maple

A022567 := proc(n)
    local x,m;
    product((1+x^m)^2,m=1..n) ;
    expand(%) ;
    coeff(%,x,n) ;
end proc: # R. J. Mathar, Jun 18 2016

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ q, q^2]^-2, {q, 0, n}]; (* Michael Somos, Jul 11 2011 *)
a[ n_] := SeriesCoefficient[ Product[ 1 + q^k, {k, n}]^2, {q, 0, n}]; (* Michael Somos, Jul 11 2011 *)
(QPochhammer[-1, x]^2/4 + O[x]^30)[[3]] (* Vladimir Reshetnikov, Sep 22 2016 *)
nmax = 50; poly = ConstantArray[0, nmax+1]; poly[[1]] = 1; poly[[2]] = 2; poly[[3]] = 1; Do[Do[Do[poly[[j+1]] += poly[[j-k+1]], {j, nmax, k, -1}]; , {p, 1, 2}], {k, 2, nmax}]; poly (* Vaclav Kotesovec, Jan 14 2017 *)

PARI

{a(n) = if( n<0, 0, polcoeff( prod( k=1, n, 1 + x^k, 1 + x * O(x^n))^2, n))}; /* Michael Somos, Mar 21 2004 */

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, Jun 03 2005 */

SageMath

# uses[EulerTransform from A166861]
b = BinaryRecurrenceSequence(0, 1, 0, 2)
a = EulerTransform(b)
print([a(n) for n in range(45)]) # Peter Luschny, Nov 11 2020

Formula

a(n) = p(n)+p(n-1)+p(n-3)+p(n-6)+...+p(n-k*(k+1)/2)+..., where p() is A000041(). E.g. a(8) = p(8)+p(7)+p(5)+p(2) = 22+15+7+2 = 46. - Vladeta Jovovic, Aug 09 2004

Expansion of q^(-1/12) * (eta(q^2) / eta(q))^2 in powers of q. - Michael Somos, Apr 27 2008

Expansion of chi(-q)^(-2) in powers of q where chi() is a Ramanujan theta function. - Michael Somos, Apr 27 2008

G.f. is a period 1 Fourier series which satisfies f(-1 / (288 t)) = (1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A022597. - Michael Somos, Apr 27 2008

G.f.: Product_{k>0} (1 + x^k)^2.

Convolution square of A000009. Convolution inverse of A022597. - Michael Somos, Apr 27 2008

Parity result: a(n) is even except when n is twice a generalized pentagonal number (i.e., of the form 2*A001318(m) for some m). - Peter Bala, Mar 19 2009

a(n) ~ exp(Pi * sqrt(2*n/3)) / (4 * 6^(1/4) * n^(3/4)) * (1 + (Pi/(12*sqrt(6)) - 3*sqrt(3/2)/(8*Pi)) / sqrt(n) + (Pi^2/1728 - 45/(256*Pi^2) - 5/64)/n). - Vaclav Kotesovec, Mar 05 2015, extended Jan 22 2017

a(0) = 1, a(n) = (2/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 03 2017

G.f.: exp(2*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018