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A058696 Number of ways to partition 2n into positive integers.

%I A058696 #57 Feb 16 2025 08:32:43
%S A058696 1,2,5,11,22,42,77,135,231,385,627,1002,1575,2436,3718,5604,8349,
%T A058696 12310,17977,26015,37338,53174,75175,105558,147273,204226,281589,
%U A058696 386155,526823,715220,966467,1300156,1741630,2323520,3087735,4087968,5392783,7089500,9289091
%N A058696 Number of ways to partition 2n into positive integers.
%C A058696 A bisection of A000041, the other one is A058695.
%C A058696 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). - _Michael Somos_, Feb 16 2014
%C A058696 a(n) is the number of partitions of 3n-2 having n as a part, for n >= 1.  Also, a(n+1) is the number of partitions of 3n having n as a part, for n >= 1.  - _Clark Kimberling_, Mar 02 2014
%H A058696 Seiichi Manyama, <a href="/A058696/b058696.txt">Table of n, a(n) for n = 0..10000</a>
%H A058696 Roland Bacher and Pierre De La Harpe, <a href="https://hal.science/hal-01285685">Conjugacy growth series of some infinitely generated groups</a>, International Mathematics Research Notices, 2016, pp.1-53. (hal-01285685v2)
%H A058696 K. Blum, <a href="https://arxiv.org/abs/2103.03196">Bounds on the Number of Graphical Partitions</a>, arXiv:2103.03196 [math.CO], 2021. See Table on p. 7.
%H A058696 Álvaro Gutiérrez and Mercedes H. Rosas, <a href="https://arxiv.org/abs/2201.00240">Partial symmetries of iterated plethysms</a>, arXiv:2201.00240 [math.CO], 2022.
%H A058696 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A058696 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A058696 Expansion of f(x^3, x^5) / f(-x)^2 in powers of x where f() is a Ramanujan theta function. - _Michael Somos_, Feb 16 2014
%F A058696 Euler transform of period 16 sequence [ 2, 2, 3, 2, 3, 1, 2, 1, 2, 1, 3, 2, 3, 2, 2, 1, ...]. - _Michael Somos_,  Apr 25 2003
%F A058696 a(n) = A000041(2*n).
%F A058696 Convolution of A000041 and A035294. - _Michael Somos_, Feb 16 2014
%F A058696 G.f.: Product_{k>=1} (1 + x^(8*k-4)) * (1 + x^(8*k)) * (1 + x^k)^2 / ((1 + x^(8*k-1)) * (1 + x^(8*k-7)) * (1 - x^k)). - _Vaclav Kotesovec_, Nov 17 2016
%F A058696 a(n) ~ exp(2*Pi*sqrt(n/3)) / (8*sqrt(3)*n). - _Vaclav Kotesovec_, Feb 16 2022
%e A058696 G.f. = 1 + 2*x + 5*x^2 + 11*x^3 + 22*x^4 + 42*x^5 + 77*x^6 + 135*x^7 + ...
%e A058696 G.f. = q^-1 + 2*q^47 + 5*q^95 + 11*q^143 + 22*q^191 + 42*q^239 + 77*q^287 + ...
%p A058696 a:= n-> combinat[numbpart](2*n):
%p A058696 seq(a(n), n=0..42);  # _Alois P. Heinz_, Jan 29 2020
%t A058696 nn=100;Table[CoefficientList[Series[Product[1/(1-x^i),{i,1,nn}],{x,0,nn}],x][[2i-1]],{i,1,nn/2}] (* _Geoffrey Critzer_, Sep 28 2013 *)
%t A058696 (* also *)
%t A058696 Table[PartitionsP[2 n], {n, 0, 40}] (* _Clark Kimberling_, Mar 02 2014 *)
%t A058696 (* also *)
%t A058696 Table[Count[IntegerPartitions[3 n - 2], p_ /; MemberQ[p, n]], {n, 20}]   (* _Clark Kimberling_, Mar 02 2014 *)
%t A058696 nmax = 60; CoefficientList[Series[Product[(1 + x^(8*k-4))*(1 + x^(8*k))*(1 + x^k)^2/((1 + x^(8*k-1))*(1 + x^(8*k-7))*(1 - x^k)), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Nov 17 2016 *)
%o A058696 (PARI) {a(n) = if( n<0, 0, polcoeff( 1 / eta(x + O(x^(2*n + 1))), 2*n))}; /* _Michael Somos_, Apr 25 2003 */
%o A058696 (PARI) a(n) = numbpart(2*n); \\ _Michel Marcus_, Sep 28 2013
%o A058696 (MuPAD) combinat::partitions::count(2*i) $i=0..54 // _Zerinvary Lajos_, Apr 16 2007
%Y A058696 Cf. A000041, A035294, A058695.
%K A058696 nonn
%O A058696 0,2
%A A058696 _N. J. A. Sloane_, Dec 31 2000