

A090867


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


12



0, 0, 1, 1, 3, 4, 6, 9, 13, 18, 23, 32, 42, 55, 69, 89, 112, 141, 175, 217, 266, 326, 396, 480, 581, 697, 834, 996, 1183, 1402, 1660, 1954, 2297, 2694, 3150, 3674, 4280, 4970, 5762, 6669, 7701, 8876, 10219, 11737, 13460, 15418, 17628, 20125, 22951, 26128, 29709
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OFFSET

0,5


COMMENTS

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
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. 12, January  June (2017).  George Beck, Apr 23 2017
a(n) is the number of partitions of n with exactly one repeated part.  Andrew Howroyd, Feb 14 2021


LINKS

George E. Andrews, Euler's Partition Identity and Two Problems of George Beck, The Mathematics Student, 86, 12:115119 (2017); Preprint.
Cristina Ballantine, Hannah E. Burson, Amanda Folsom, ChiYun Hsu, Isabella Negrini and Boya Wen, On a Partition Identity of Lehmer, arXiv:2109.00609 [math.CO], 2021.


FORMULA

G.f.: Sum_{m>0} x^(2*m)/(1x^(2*m))/Product_{m>0} (1x^(2*m1)).
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 EulerMascheroni constant A001620.  Vaclav Kotesovec, May 25 2018


MAPLE

b:= proc(n, i, t) option remember; `if`(n=0, t, `if`(i<1, 0,
b(n, i1, t)+`if`(i>n or t=1 and i::even, 0,
add(b(ni*j, i1, `if`(i::even, 1, t)), j=1..n/i))))
end:
a:= n> b(n$2, 0):
add(numtheory[tau](k)*A000009(n2*k), k=1..n/2) ;


MATHEMATICA

f[n_] := Count[ Plus @@@ Mod[ Union /@ IntegerPartitions[n] + 1, 2], 1]; Table[ f[n], {n, 0, 50}] (* Robert G. Wilson v, Feb 16 2004 *)
a[n_] := Sum[DivisorSigma[0, k] PartitionsQ[n2k], {k, 1, n/2}];


PROG

(PARI) seq(n)={Vec(sum(k=1, n\2, x^(2*k)/(1x^(2*k)) + O(x*x^n))/prod(k=1, n\2, 1x^(2*k1) + O(x*x^n)), (n+1))} \\ Andrew Howroyd, Feb 13 2021


CROSSREFS



KEYWORD

easy,nonn


AUTHOR



EXTENSIONS



STATUS

approved



