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A265251 Number of partitions of n such that there is exactly one part which occurs three times, while all other parts occur only once.

%I A265251 #104 Mar 28 2025 11:28:43
%S A265251 0,0,0,1,0,1,2,2,2,4,6,6,9,10,14,19,22,26,35,40,50,63,74,88,107,127,
%T A265251 150,181,213,249,296,345,401,473,546,636,741,853,983,1138,1306,1498,
%U A265251 1722,1967,2247,2574,2925,3327,3788,4294,4866,5516,6233,7036,7947,8953
%N A265251 Number of partitions of n such that there is exactly one part which occurs three times, while all other parts occur only once.
%C A265251 Conjecture: a(n) is also the difference between the number of parts in the distinct partitions of n and the number of distinct parts in the odd partitions of n (offset 0). For example, if n = 5, there are 5 parts in the distinct partitions of 5 (5, 41, 32) and 4 distinct parts in the odd partitions of 5 (namely, 5,3,1,1 in 5,311,11111) with difference 1. - _George Beck_, Apr 22 2017
%C A265251 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
%H A265251 Vaclav Kotesovec, <a href="/A265251/b265251.txt">Table of n, a(n) for n = 0..15866</a> (terms 0..10000 from Alois P. Heinz)
%H A265251 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 A265251 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 A265251 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 A265251 Cristina Ballantine, Hannah Burson, William Craig, Amanda Folsom, and Boya Wen, <a href="https://arxiv.org/abs/2303.16512">Hook length biases and general linear partition inequalities</a>, arXiv:2303.16512 [math.CO], 2023.
%H A265251 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 A265251 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 A265251 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 A265251 Cristina Ballantine, Hannah Burson, William Craig, Amanda Folsom, and Boya Wen, <a href="https://www.mat.univie.ac.at/~slc/wpapers/FPSAC2023/39.html">Hook length bias in odd versus distinct partitions</a>, Séminaire Lotharingien de Combinatoire, Proc. 35th Conf. Formal Power Series and Alg. Comb. (Davis, 2023) Vol. 89B, Art. #39.
%H A265251 Alexander Berkovich and Aritram Dhar, <a href="https://arxiv.org/abs/2503.17571">Finite analogs of partition bias related to hook length two and a variant of Sylvester's map</a>, arXiv:2503.17571 [math.CO], 2025. See p. 19.
%H A265251 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 A265251 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 A265251 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.
%F A265251 G.f.: Sum_{k>=1} x^{3k}/(1+x^k)*Product_{i>=1} (1+x^i).
%F A265251 a(n) ~ c * exp(Pi*sqrt(n/3)) / n^(1/4), where c = 3^(1/4) * (2*log(2) - 1) / (4*Pi) = 0.040456547528... - _Vaclav Kotesovec_, May 24 2018
%e A265251 a(9) = 4 because we have [2,2,2,3], [3,3,3], [1,1,1,2,4], and [1,1,1,6].
%p A265251 g := add(x^(3*k)/(1+x^k), k = 1 .. 100)*mul(1+x^i, i = 1 .. 100): gser := series(g, x = 0, 80): seq(coeff(gser, x, m), m = 0 .. 75);
%p A265251 # second Maple program:
%p A265251 b:= proc(n, i, t) option remember; `if`(n>i*(i+5-4*t)/2, 0,
%p A265251      `if`(n=0, t, b(n, i-1, t)+`if`(i>n, 0, b(n-i, i-1, t)+
%p A265251      `if`(t=1 or 3*i>n, 0, b(n-3*i, i-1, 1)))))
%p A265251     end:
%p A265251 a:= n-> b(n$2, 0):
%p A265251 seq(a(n), n=0..100);  # _Alois P. Heinz_, Dec 28 2015
%t A265251 b[n_, i_, t_] := b[n, i, t] = If[n > i*(i + 5 - 4*t)/2, 0, If[n == 0, t, b[n, i - 1, t] + If[i > n, 0, b[n - i, i - 1, t] + If[t == 1 || 3*i > n, 0, b[n - 3*i, i - 1, 1]]]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Dec 11 2016, after _Alois P. Heinz_ *)
%t A265251 Take[ CoefficientList[ Expand[ Sum[x^(3k)/(1 + x^k), {k, 60}] Product[1 + x^i, {i, 60}]], x], 60] (* slower than above *) (* _Robert G. Wilson v_, Apr 24 2017 *)
%o A265251 (PARI) x='x + O('x^54); concat([0, 0, 0],Vec(sum(k=1, 54, x^(3*k)/(1 + x^k)* prod(i=1, 54, 1 + x^i)))) \\ _Indranil Ghosh_, Apr 24 2017
%Y A265251 Column k=3 of A266477.
%Y A265251 Cf. A015723, A090858, A090867.
%K A265251 nonn
%O A265251 0,7
%A A265251 _Emeric Deutsch_, Dec 28 2015