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

%I A022567 #142 Feb 16 2025 08:32:34
%S A022567 1,2,3,6,9,14,22,32,46,66,93,128,176,238,319,426,562,736,960,1242,
%T A022567 1598,2048,2608,3306,4175,5248,6570,8198,10190,12622,15589,19190,
%U A022567 23552,28830,35190,42842,52034,63040,76198,91904,110604,132832,159216,190464,227417
%N A022567 Expansion of Product_{m>=1} (1+x^m)^2.
%C A022567 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A022567 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
%C A022567 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
%C A022567 Euler transform of period 2 sequence [2,0,2,0,...]. - _Emeric Deutsch_, Mar 22 2005
%C A022567 Equals A000041 convolved with A010054. - _Gary W. Adamson_, Jun 11 2009
%C A022567 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
%C A022567 Number of 2-regular bipartitions of n. - _N. J. A. Sloane_, Oct 20 2019
%C A022567 The least gap is also known as the minimal excludant or mex; see Andrews and Newman. - _George Beck_, Dec 10 2020
%D A022567 P. J. Grabner, A. Knopfmacher, Analysis of some new partition statistics, Ramanujan J., 12, 2006, 439-454.
%D A022567 Kathiravan, T., and S. N. Fathima. "On L-regular bipartitions modulo L." The Ramanujan Journal 44.3 (2017): 549-558.
%H A022567 Alois P. Heinz, <a href="/A022567/b022567.txt">Table of n, a(n) for n = 0..10000</a>
%H A022567 George E. Andrews, David Newman, <a href="https://doi.org/10.1007/s00026-019-00427-w">Partitions and the Minimal Excludant</a>, Annals of Combinatorics, Volume 23, May 2019, Pages 249-254.
%H A022567 Cristina Ballantine, Mircea Merca, <a href="https://arxiv.org/abs/1710.05960">Bisected theta series, least r-gaps in partitions, and polygonal numbers</a>, arXiv:1710.05960 [math.CO], 2017.
%H A022567 Helena Bergold, Lukas Egeling, and Hung. P. Hoang, <a href="https://arxiv.org/abs/2411.19208">Signotopes with few plus signs</a>, arXiv:2411.19208 [math.CO], 2024. See p. 14.
%H A022567 J. Currie, N. Rampersad, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Currie/currie12.html">Binary words avoiding xx^Rx and strongly unimodal sequences</a>, JIS 18 (2015) #15.10.3.
%H A022567 Alejandro Erickson, Frank Ruskey, <a href="http://arxiv.org/abs/1304.0070">Enumerating maximal tatami mat coverings of square grids with v vertical dominoes</a>, arXiv:1304.0070 [math.CO], 2013.
%H A022567 Alejandro Erickson and Mark Schurch, <a href="http://arxiv.org/abs/1110.5103">Monomer-dimer tatami tilings of square regions</a>, arXiv preprint arXiv:1110.5103 [math.CO], 2011.
%H A022567 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=852">Encyclopedia of Combinatorial Structures 852</a>
%H A022567 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], Sep 30 2015, p. 8.
%H A022567 Mircea Merca, <a href="http://dx.doi.org/10.1016/j.jnt.2014.10.009">A new look on the generating function for the number of divisors</a>, Journal of Number Theory, Volume 149, April 2015, Pages 57-69.
%H A022567 Mircea Merca, <a href="http://dx.doi.org/10.1016/j.jnt.2015.08.014">Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer</a>, Journal of Number Theory, Volume 160, March 2016, Pages 60-75. See q(n)'.
%H A022567 Mbavhalelo Mulokwe and Konstantinos Zoubos, <a href="https://arxiv.org/abs/2403.08531">Free fermions, neutrality and modular transformations</a>, arXiv:2403.08531 [hep-th], 2024.
%H A022567 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A022567 Jacob Sprittulla, <a href="https://arxiv.org/abs/2008.09984">On Colored Factorizations</a>, arXiv:2008.09984 [math.CO], 2020.
%H A022567 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A022567 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
%F A022567 Expansion of q^(-1/12) * (eta(q^2) / eta(q))^2 in powers of q. - _Michael Somos_, Apr 27 2008
%F A022567 Expansion of chi(-q)^(-2) in powers of q where chi() is a Ramanujan theta function. - _Michael Somos_, Apr 27 2008
%F A022567 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
%F A022567 G.f.: Product_{k>0} (1 + x^k)^2.
%F A022567 Convolution square of A000009. Convolution inverse of A022597. - _Michael Somos_, Apr 27 2008
%F A022567 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
%F A022567 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
%F A022567 a(0) = 1, a(n) = (2/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - _Seiichi Manyama_, Apr 03 2017
%F A022567 G.f.: exp(2*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))). - _Ilya Gutkovskiy_, Feb 06 2018
%e A022567 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 + ...
%e A022567 G.f. = q + 2*q^13 + 3*q^25 + 6*q^37 + 9*q^49 + 14*q^61 + 22*q^73 + 32*q^85 + ...
%p A022567 A022567 := proc(n)
%p A022567     local x,m;
%p A022567     product((1+x^m)^2,m=1..n) ;
%p A022567     expand(%) ;
%p A022567     coeff(%,x,n) ;
%p A022567 end proc: # _R. J. Mathar_, Jun 18 2016
%t A022567 a[ n_] := SeriesCoefficient[ QPochhammer[ q, q^2]^-2, {q, 0, n}]; (* _Michael Somos_, Jul 11 2011 *)
%t A022567 a[ n_] := SeriesCoefficient[ Product[ 1 + q^k, {k, n}]^2, {q, 0, n}]; (* _Michael Somos_, Jul 11 2011 *)
%t A022567 (QPochhammer[-1, x]^2/4 + O[x]^30)[[3]] (* _Vladimir Reshetnikov_, Sep 22 2016 *)
%t A022567 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 *)
%o A022567 (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 */
%o A022567 (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 */
%o A022567 (Magma) Coefficients(&*[(1+x^m)^2:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // _G. C. Greubel_, Feb 26 2018
%o A022567 (SageMath) # uses[EulerTransform from A166861]
%o A022567 b = BinaryRecurrenceSequence(0, 1, 0, 2)
%o A022567 a = EulerTransform(b)
%o A022567 print([a(n) for n in range(45)]) # _Peter Luschny_, Nov 11 2020
%Y A022567 Cf. A000009, A022597, A285221, A304048.
%Y A022567 Cf. A010054. - _Gary W. Adamson_, Jun 11 2009
%Y A022567 Column k=2 of A286335.
%Y A022567 Number of r-regular bipartitions of n for r = 2,3,4,5,6: A022567, A328547, A001936, A263002, A328548.
%K A022567 nonn
%O A022567 0,2
%A A022567 _N. J. A. Sloane_, Jun 14 1998