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%I #118 Dec 18 2023 08:25:17
%S 0,0,1,1,3,4,6,9,13,18,23,32,42,55,69,89,112,141,175,217,266,326,396,
%T 480,581,697,834,996,1183,1402,1660,1954,2297,2694,3150,3674,4280,
%U 4970,5762,6669,7701,8876,10219,11737,13460,15418,17628,20125,22951,26128,29709
%N Number of partitions of n such that the set of even parts has only one element.
%C Conjecture: a(n) is also the difference between the number of parts in the odd partitions of n and the number of parts in the distinct partitions of n (offset 0). For example, if n = 5, there are 9 parts in the odd partitions of 5 (5, 311, 11111) and 5 parts in the distinct partitions of 5 (5, 41, 32), with difference 4. - _George Beck_, Apr 22 2017
%C George E. Andrews has kindly informed me that he has proved this conjecture and the result will be included in his article "Euler's Partition Identity and Two Problems of George Beck" which will appear in The Mathematics Student, 86, Nos. 1-2, January - June (2017). - _George Beck_, Apr 23 2017
%C a(n) is the number of partitions of n with exactly one repeated part. - _Andrew Howroyd_, Feb 14 2021
%H Vaclav Kotesovec, <a href="/A090867/b090867.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..5000 from Alois P. Heinz)
%H Tewodros Amdeberhan, George E. Andrews, and Cristina Ballantine, <a href="https://arxiv.org/abs/2204.00105">Refinements of Beck-type partition identities</a>, arXiv:2204.00105 [math.CO], 2022.
%H George E. Andrews, Euler's Partition Identity and Two Problems of George Beck, The Mathematics Student, 86, 1-2:115-119 (2017); <a href="https://georgeandrews1.github.io/pdf/326.pdf">Preprint</a>.
%H Cristina Ballantine and Richard Bielak, <a href="https://arxiv.org/abs/1803.06394">Combinatorial proofs of two Euler type identities due to Andrews</a>, arXiv:1803.06394 [math.CO], 2018.
%H Cristina Ballantine and Amanda Welch, <a href="https://arxiv.org/abs/2006.02335">Beck-type identities for Euler pairs of order r</a>, arXiv:2006.02335 [math.NT], 2020.
%H Cristina Ballantine and Amanda Welch, <a href="https://arxiv.org/abs/2011.08220">Beck-type identities: new combinatorial proofs and a theorem for parts congruent to t mod r</a>, arXiv:2011.08220 [math.CO], 2020.
%H Cristina Ballantine and Amanda Welch, <a href="https://arxiv.org/abs/2101.06260">Beck-type companion identities for Franklin's identity</a>, arXiv:2101.06260 [math.CO], 2021.
%H Cristina Ballantine and Amanda Welch, <a href="https://doi.org/10.1007/s11139-021-00469-w">Beck-type identities: new combinatorial proofs and a modular refinement</a>, Ramanujan J. (2021).
%H Cristina Ballantine and Mircea Merca, <a href="https://doi.org/10.1007/s11139-019-00184-7">Combinatorial proofs of two theorems related to the number of even parts in all partitions of n into distinct parts</a>, Ramanujan J., 54:1 (2021), 107-112.
%H Cristina Ballantine, Hannah E. Burson, Amanda Folsom, Chi-Yun Hsu, Isabella Negrini and Boya Wen, <a href="https://arxiv.org/abs/2109.00609">On a Partition Identity of Lehmer</a>, arXiv:2109.00609 [math.CO], 2021.
%H Cristina Ballantine and Amanda Folsom, <a href="https://arxiv.org/abs/2303.03330">On the number of parts in all partitions enumerated by the Rogers-Ramanujan identities</a>, arXiv:2303.03330 [math.NT], 2023.
%H Shishuo Fu and Dazhao Tang, <a href="https://arxiv.org/abs/1705.05046">Generalizing a partition theorem of Andrews</a>, arXiv:1705.05046 [math.CO], 2017.
%H Jia Huang, <a href="https://arxiv.org/abs/1812.11010">Compositions with restricted parts</a>, arXiv:1812.11010 [math.CO], 2018. Also Discrete Masth., 343 (2020), # 111875.
%H Runqiao Li and Andrew Y. Z. Wang, <a href="https://doi.org/10.1007/s11139-021-00468-x">The dual form of Beck type identities</a>, Ramanujan J. (2021).
%H 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.
%H Mircea Merca, <a href="https://arxiv.org/abs/2005.03619">On the partitions into distinct parts and odd parts</a>, arXiv:2005.03619 [math.CO], 2020.
%H Aritro Pathak, <a href="https://arxiv.org/abs/2004.03596">On certain partition bijections related to Euler's partition problem</a>, arXiv:2004.03596 [math.CO], 2020. Also Discrete Mathematics 345.2 (2022): 112673.
%H Jane Y. X. Yang, <a href="https://arxiv.org/abs/1801.06815">Combinatorial proofs and generalizations on conjectures related with Euler's partition theorem</a>, arXiv:1801.06815 [math.CO], 2018.
%F G.f.: Sum_{m>0} x^(2*m)/(1-x^(2*m))/Product_{m>0} (1-x^(2*m-1)).
%F a(n) ~ 3^(1/4) * (2*gamma + log(3*n/Pi^2)) * exp(Pi*sqrt(n/3)) / (8*Pi*n^(1/4)), where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, May 25 2018
%F a(n) = A341494(n) + A341495(n) = A341496(n) + A341497(n). - _Andrew Howroyd_, Feb 14 2021
%p b:= proc(n, i, t) option remember; `if`(n=0, t, `if`(i<1, 0,
%p b(n, i-1, t)+`if`(i>n or t=1 and i::even, 0,
%p add(b(n-i*j, i-1, `if`(i::even, 1, t)), j=1..n/i))))
%p end:
%p a:= n-> b(n$2, 0):
%p seq(a(n), n=0..70); # _Alois P. Heinz_, Jun 17 2016
%p A090867 := proc(n)
%p add(numtheory[tau](k)*A000009(n-2*k),k=1..n/2) ;
%p end proc: # _R. J. Mathar_, Jun 18 2016
%t f[n_] := Count[ Plus @@@ Mod[ Union /@ IntegerPartitions[n] + 1, 2], 1]; Table[ f[n], {n, 0, 50}] (* _Robert G. Wilson v_, Feb 16 2004 *)
%t a[n_] := Sum[DivisorSigma[0, k] PartitionsQ[n-2k], {k, 1, n/2}];
%t a /@ Range[0, 70] (* _Jean-François Alcover_, May 24 2021, after _R. J. Mathar_ *)
%o (PARI) seq(n)={Vec(sum(k=1, n\2, x^(2*k)/(1-x^(2*k)) + O(x*x^n))/prod(k=1, n\2, 1-x^(2*k-1) + O(x*x^n)), -(n+1))} \\ _Andrew Howroyd_, Feb 13 2021
%Y Cf. A038348, A265251, A341494, A341495, A341496, A341497.
%K easy,nonn
%O 0,5
%A _Vladeta Jovovic_, Feb 12 2004
%E More terms from _Robert G. Wilson v_, Feb 16 2004
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