cp's OEIS Frontend

A090867 Number of partitions of n such that the set of even parts has only one element.

%I A090867 #134 Jul 23 2025 10:04:11
%S A090867 0,0,1,1,3,4,6,9,13,18,23,32,42,55,69,89,112,141,175,217,266,326,396,
%T A090867 480,581,697,834,996,1183,1402,1660,1954,2297,2694,3150,3674,4280,
%U A090867 4970,5762,6669,7701,8876,10219,11737,13460,15418,17628,20125,22951,26128,29709
%N A090867 Number of partitions of n such that the set of even parts has only one element.
%C A090867 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 A090867 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 A090867 a(n) is the number of partitions of n with exactly one repeated part. - _Andrew Howroyd_, Feb 14 2021
%H A090867 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 A090867 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 A090867 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 A090867 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 A090867 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 A090867 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 A090867 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 A090867 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 A090867 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 A090867 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 A090867 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 A090867 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 A090867 Gabriel Gray, Emily Payne, and Ren Watson, <a href="https://math.oregonstate.edu/sites/math.oregonstate.edu/files/2024-08/EmilyGabeRen_0.pdf">Generalized partition identities and fixed perimeter analogues</a>, Oregon State Univ. (2024). See pp. 2, 49.
%H A090867 Gabriel Gray, David Hovey, Brandt Kronholm, Emily Payne, Holly Swisher, and Ren Watson, <a href="https://arxiv.org/abs/2410.17378">A generalization of Franklin's partition identity and a Beck-type companion identity</a>, arXiv:2410.17378 [math.NT], 2024.  See p. 12. See also <a href="https://doi.org/10.1007/s11139-025-01121-7">Ramanujan J.</a> (2025) Volume 67, Art. No. 100.
%H A090867 Gabriel Gray, Emily Payne, Holly Swisher, and Ren Watson, <a href="https://arxiv.org/abs/2502.12394">Fixed perimeter analogues of some partition results</a>, arXiv:2502.12394 [math.CO], 2025. See p. 15.
%H A090867 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 A090867 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 A090867 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 A090867 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 A090867 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 A090867 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 A090867 G.f.: Sum_{m>0} x^(2*m)/(1-x^(2*m))/Product_{m>0} (1-x^(2*m-1)).
%F A090867 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 A090867 a(n) = A341494(n) + A341495(n) = A341496(n) + A341497(n). - _Andrew Howroyd_, Feb 14 2021
%p A090867 b:= proc(n, i, t) option remember; `if`(n=0, t, `if`(i<1, 0,
%p A090867       b(n, i-1, t)+`if`(i>n or t=1 and i::even, 0,
%p A090867       add(b(n-i*j, i-1, `if`(i::even, 1, t)), j=1..n/i))))
%p A090867     end:
%p A090867 a:= n-> b(n$2, 0):
%p A090867 seq(a(n), n=0..70);  # _Alois P. Heinz_, Jun 17 2016
%p A090867 A090867 := proc(n)
%p A090867     add(numtheory[tau](k)*A000009(n-2*k),k=1..n/2) ;
%p A090867 end proc: # _R. J. Mathar_, Jun 18 2016
%t A090867 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 A090867 a[n_] := Sum[DivisorSigma[0, k] PartitionsQ[n-2k], {k, 1, n/2}];
%t A090867 a /@ Range[0, 70] (* _Jean-François Alcover_, May 24 2021, after _R. J. Mathar_ *)
%o A090867 (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 A090867 Cf. A038348, A265251, A341494, A341495, A341496, A341497.
%K A090867 easy,nonn
%O A090867 0,5
%A A090867 _Vladeta Jovovic_, Feb 12 2004
%E A090867 More terms from _Robert G. Wilson v_, Feb 16 2004