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A265251
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Number of partitions of n such that there is exactly one part which occurs three times, while all other parts occur only once.
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4
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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
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OFFSET
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0,7
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COMMENTS
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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
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
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LINKS
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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, 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.
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FORMULA
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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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EXAMPLE
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a(9) = 4 because we have [2,2,2,3], [3,3,3], [1,1,1,2,4], and [1,1,1,6].
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MAPLE
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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):
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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