A341494 -id:A341494 - cp's OEIS frontend

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

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

Offset: 0

Vladeta Jovovic, Feb 12 2004

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. 1-2, 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

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..5000 from Alois P. Heinz)
Tewodros Amdeberhan, George E. Andrews, and Cristina Ballantine, Refinements of Beck-type partition identities, arXiv:2204.00105 [math.CO], 2022.
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 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 and Mircea Merca, Combinatorial proofs of two theorems related to the number of even parts in all partitions of n into distinct parts, Ramanujan J., 54:1 (2021), 107-112.
Cristina Ballantine, Hannah E. Burson, Amanda Folsom, Chi-Yun Hsu, Isabella Negrini and Boya Wen, On a Partition Identity of Lehmer, arXiv:2109.00609 [math.CO], 2021.
Cristina Ballantine and Amanda Folsom, On the number of parts in all partitions enumerated by the Rogers-Ramanujan identities, arXiv:2303.03330 [math.NT], 2023.
Shishuo Fu and Dazhao Tang, Generalizing a partition theorem of Andrews, arXiv:1705.05046 [math.CO], 2017.
Gabriel Gray, Emily Payne, and Ren Watson, Generalized partition identities and fixed perimeter analogues, Oregon State Univ. (2024). See pp. 2, 49.
Gabriel Gray, David Hovey, Brandt Kronholm, Emily Payne, Holly Swisher, and Ren Watson, A generalization of Franklin's partition identity and a Beck-type companion identity, arXiv:2410.17378 [math.NT], 2024. See p. 12. See also Ramanujan J. (2025) Volume 67, Art. No. 100.
Gabriel Gray, Emily Payne, Holly Swisher, and Ren Watson, Fixed perimeter analogues of some partition results, arXiv:2502.12394 [math.CO], 2025. See p. 15.
Jia Huang, Compositions with restricted parts, arXiv:1812.11010 [math.CO], 2018. Also Discrete Masth., 343 (2020), # 111875.
Runqiao Li and Andrew Y. Z. Wang, The dual form of Beck type identities, Ramanujan J. (2021).
Mircea Merca, Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer, Journal of Number Theory, Volume 160, March 2016, Pages 60-75.
Mircea Merca, On the partitions into distinct parts and odd parts, arXiv:2005.03619 [math.CO], 2020.
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.
Jane Y. X. Yang, Combinatorial proofs and generalizations on conjectures related with Euler's partition theorem, arXiv:1801.06815 [math.CO], 2018.

Cf. A038348, A265251, A341494, A341495, A341496, A341497.

b:= proc(n, i, t) option remember; `if`(n=0, t, `if`(i<1, 0,
      b(n, i-1, t)+`if`(i>n or t=1 and i::even, 0,
      add(b(n-i*j, i-1, `if`(i::even, 1, t)), j=1..n/i))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..70);  # Alois P. Heinz, Jun 17 2016
A090867 := proc(n)
    add(numtheory[tau](k)*A000009(n-2*k),k=1..n/2) ;
end proc: # R. J. Mathar, Jun 18 2016

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[n-2k], {k, 1, n/2}];
a /@ Range[0, 70] (* Jean-François Alcover, May 24 2021, after R. J. Mathar *)

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

G.f.: Sum_{m>0} x^(2*m)/(1-x^(2*m))/Product_{m>0} (1-x^(2*m-1)).
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
a(n) = A341494(n) + A341495(n) = A341496(n) + A341497(n). - Andrew Howroyd, Feb 14 2021

More terms from Robert G. Wilson v, Feb 16 2004

A341495 Number of partitions of n into an odd number of parts such that the set of even parts has only one element.

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 5, 2, 9, 5, 17, 9, 30, 16, 49, 26, 78, 43, 122, 67, 184, 101, 272, 151, 397, 222, 567, 320, 802, 454, 1121, 637, 1545, 884, 2112, 1214, 2863, 1651, 3842, 2227, 5123, 2979, 6782, 3957, 8913, 5218, 11648, 6840, 15136, 8914, 19555, 11552, 25143

Offset: 0

Andrew Howroyd, Feb 13 2021

nonn

			The a(2) = 1 partition is: 2.
The a(4) = 2 partitions are: 4, 1+1+2.
The a(5) = 1 partition is: 1+2+2.
The a(6) = 5 partitions are: 6, 1+1+4, 1+2+3, 2+2+2, 1+1+1+1+2.

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Cristina Ballantine and Mircea Merca, Combinatorial proofs of two theorems related to the number of even parts in all partitions of n into distinct parts, Ramanujan J., 54:1 (2021), 107-112.

Cf. A090867, A341494, A341496, A341497.

P[n_, c_] := c*Sum[x^(2k)/(1 - c*x^(2k)) + O[x]^n, {k, 1, n/2}]/
     Product[1 - c*x^(2k - 1) + O[x]^n, {k, 1, n/2}];
CoefficientList[(P[100, 1] - P[100, -1])/2, x] (* Jean-François Alcover, May 24 2021, from PARI code *)

P(n,c)={c*sum(k=1, n\2, x^(2*k)/(1-c*x^(2*k)) + O(x*x^n))/prod(k=1, n\2, 1-c*x^(2*k-1) + O(x*x^n))}
seq(n)={Vec(P(n,1) - P(n,-1), -(n+1))/2}

G.f.: (P(x,1) - P(x,-1))/2 where P(x,c) = (Sum_{k>=1} c*x^(2*k)/(1-c*x^(2*k))) / (Product_{k>=1} 1-c*x^(2*k-1)).

a(n) = A090867(n) - A341494(n).

A341496 Number of partitions of n with exactly one repeated part and that part is even.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 2, 4, 5, 6, 9, 12, 16, 20, 26, 34, 43, 53, 67, 82, 101, 124, 151, 184, 222, 267, 320, 381, 454, 539, 637, 752, 884, 1038, 1214, 1417, 1651, 1920, 2227, 2578, 2979, 3437, 3957, 4547, 5218, 5980, 6840, 7815, 8914, 10154, 11552, 13122

Offset: 0

Andrew Howroyd, Feb 13 2021

nonn

			The a(4) = 1 partition is: 2+2.
The a(5) = 1 partition is: 1+2+2.
The a(6) = 1 partition is: 2+2+2.
The a(7) = 2 partitions are: 2+2+3, 1+2+2+2.
The a(8) = 4 partitions are: 4+4, 2+2+4, 1+2+2+3, 2+2+2+2.

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Cristina Ballantine and Mircea Merca, Combinatorial proofs of two theorems related to the number of even parts in all partitions of n into distinct parts, Ramanujan J., 54:1 (2021), 107-112.

Cf. A090867, A116680, A341494, A341495, A341497.

seq(n)={Vec(sum(k=1, n\4, x^(4*k)/(1 - x^(4*k)) + O(x*x^n)) * prod(k=1, n, 1 + x^k + O(x*x^n)), -(n+1))}

G.f.: (Sum_{k>=1} x^(4*k)/(1 - x^(4*k))) * Product_{k>=1} (1 + x^k).
a(n) = A090867(n) - A341497(n).
a(n) = A341497(n) - A116680(n).
a(n) = A341494(n) for even n; a(n) = A341495(n) for odd n.

A341497 Number of partitions of n with exactly one repeated part and that part is odd.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 5, 7, 9, 13, 17, 23, 30, 39, 49, 63, 78, 98, 122, 150, 184, 225, 272, 329, 397, 475, 567, 676, 802, 948, 1121, 1317, 1545, 1810, 2112, 2460, 2863, 3319, 3842, 4442, 5123, 5897, 6782, 7780, 8913, 10200, 11648, 13285, 15136, 17214, 19555, 22191, 25143

Offset: 0

Andrew Howroyd, Feb 13 2021

			The a(2) = 1 partition is: 1+1.
The a(3) = 1 partition is: 1+1+1.
The a(4) = 2 partitions are: 1+1+2, 1+1+1+1.
The a(5) = 3 partitions are: 1+1+3, 1+1+1+2, 1+1+1+1+1.

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Cristina Ballantine and Mircea Merca, Combinatorial proofs of two theorems related to the number of even parts in all partitions of n into distinct parts, Ramanujan J., 54:1 (2021), 107-112.

Cf. A067588, A090867, A116676, A116680, A341494, A341495, A341496.

seq(n)={Vec(sum(k=1, (n+2)\4, x^(4*k-2)/(1 - x^(4*k-2)) + O(x*x^n)) * prod(k=1, n, 1 + x^k + O(x*x^n)), -(n+1))}

G.f.: (Sum_{k>=1} x^(4*k-2)/(1 - x^(4*k-2))) * Product_{k>=1} (1 + x^k).
a(n) = A090867(n) - A341496(n).
a(n) = A116680(n) + A341496(n).
a(n) = A341495(n) for even n; a(n) = A341494(n) for odd n.
a(n) = (A067588(n) - A116676(n))/2. - Peter Bala, Jan 13 2025

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

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A341495 Number of partitions of n into an odd number of parts such that the set of even parts has only one element.

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A341496 Number of partitions of n with exactly one repeated part and that part is even.

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A341497 Number of partitions of n with exactly one repeated part and that part is odd.

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