A265251 Number of partitions of n such that there is exactly one part which occurs three times, while all other parts occur only once.
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, 150, 181, 213, 249, 296, 345, 401, 473, 546, 636, 741, 853, 983, 1138, 1306, 1498, 1722, 1967, 2247, 2574, 2925, 3327, 3788, 4294, 4866, 5516, 6233, 7036, 7947, 8953
Offset: 0
Keywords
Examples
a(9) = 4 because we have [2,2,2,3], [3,3,3], [1,1,1,2,4], and [1,1,1,6].
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..15866 (terms 0..10000 from Alois P. Heinz)
- George E. Andrews, Euler's Partition Identity and Two Problems of George Beck, The Mathematics Student, 86, 1-2:115-119 (2017); Preprint.
- Cristina Ballantine and Richard Bielak, Combinatorial proofs of two Euler type identities due to Andrews, arXiv:1803.06394 [math.CO], 2018.
- Cristina Ballantine and Amanda Welch, Beck-type identities for Euler pairs of order $r$, arXiv:2006.02335 [math.NT], 2020.
- Cristina Ballantine, Hannah Burson, William Craig, Amanda Folsom, and Boya Wen, Hook length biases and general linear partition inequalities, arXiv:2303.16512 [math.CO], 2023.
- Cristina Ballantine and Amanda Welch, Beck-type identities: new combinatorial proofs and a theorem for parts congruent to t mod r, arXiv:2011.08220 [math.CO], 2020.
- Cristina Ballantine and Amanda Welch, Beck-type companion identities for Franklin's identity, arXiv:2101.06260 [math.CO], 2021.
- Cristina Ballantine and Amanda Welch, Beck-type identities: new combinatorial proofs and a modular refinement, Ramanujan J. (2021).
- Cristina Ballantine, Hannah Burson, William Craig, Amanda Folsom, and Boya Wen, Hook length bias in odd versus distinct partitions, Séminaire Lotharingien de Combinatoire, Proc. 35th Conf. Formal Power Series and Alg. Comb. (Davis, 2023) Vol. 89B, Art. #39.
- Alexander Berkovich and Aritram Dhar, Finite analogs of partition bias related to hook length two and a variant of Sylvester's map, arXiv:2503.17571 [math.CO], 2025. See p. 19.
- Shishuo Fu and Dazhao Tang, Generalizing a partition theorem of Andrews, arXiv:1705.05046 [math.CO], 2017.
- Runqiao Li and Andrew Y. Z. Wang, The dual form of Beck type identities, Ramanujan J. (2021).
- Aritro Pathak, On certain partition bijections related to Euler's partition problem, arXiv:2004.03596 [math.CO], 2020. Also Discrete Mathematics 345.2 (2022): 112673.
Programs
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Maple
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); # second Maple program: b:= proc(n, i, t) option remember; `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 or 3*i>n, 0, b(n-3*i, i-1, 1))))) end: a:= n-> b(n$2, 0): seq(a(n), n=0..100); # Alois P. Heinz, Dec 28 2015 -
Mathematica
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 *) 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 *) -
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
Formula
G.f.: Sum_{k>=1} x^{3k}/(1+x^k)*Product_{i>=1} (1+x^i).
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
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