A090867 -id:A090867 - cp's OEIS frontend

A038348 Expansion of (1/(1-x^2))*Product_{m>=0} 1/(1-x^(2m+1)).

1, 1, 2, 3, 4, 6, 8, 11, 14, 19, 24, 31, 39, 49, 61, 76, 93, 114, 139, 168, 203, 244, 292, 348, 414, 490, 579, 682, 801, 938, 1097, 1278, 1487, 1726, 1999, 2311, 2667, 3071, 3531, 4053, 4644, 5313, 6070, 6923, 7886, 8971, 10190, 11561

Offset: 0

N. J. A. Sloane

nonn

Number of partitions of n+2 with exactly one even part. - Vladeta Jovovic, Sep 10 2003
Also, number of partitions of n with at most one even part. - Vladeta Jovovic, Sep 10 2003
Also total number of parts, counted without multiplicity, in all partitions of n into odd parts, offset 1. - Vladeta Jovovic, Mar 27 2005
a(n) = Sum_{k>=1} k*A116674(n+1,k). - Emeric Deutsch, Feb 22 2006
Equals row sums of triangle A173305. - Gary W. Adamson, Feb 15 2010
Equals partial sums of A025147 (observed by Jonathan Vos Post, proved by several correspondents).
Conjecture: The n-th derivative of Gamma(x+1) at x = 0 has a(n+1) terms. For example, d^4/dx^4_(x = 0) Gamma(x+1) = 8*eulergamma*zeta(3) + eulergamma^4 + eulergamma^2*Pi^2 + 3*Pi^4/20 which has a(5) = 4 terms. - David Ulgenes, Dec 05 2023

			From _Gus Wiseman_, Sep 23 2019: (Start)
Also the number of integer partitions of n that are strict except possibly for any number of 1's. For example, the a(1) = 1 through a(7) = 11 partitions are:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (21)   (31)    (32)     (42)      (43)
             (111)  (211)   (41)     (51)      (52)
                    (1111)  (311)    (321)     (61)
                            (2111)   (411)     (421)
                            (11111)  (3111)    (511)
                                     (21111)   (3211)
                                     (111111)  (4111)
                                               (31111)
                                               (211111)
                                               (1111111)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Cristina Ballantine and Mircea Merca, On identities of Watson type, Ars Mathematica Contemporanea (2019) Vol. 17, 277-290.
Kevin Beanland and Hung Viet Chu, On Schreier-type Sets, Partitions, and Compositions, arXiv:2311.01926 [math.CO], 2023.
P. Flajolet and B. Salvy, Euler sums and contour integral representations, Experimental Mathematics, Vol. 7 Issue 1 (1998).
J. Fulman, Random matrix theory over finite fields, Bull. Amer. Math. Soc. (N.S.), 39 (2002), no. 1, 51--85. MR1864086 (2002i:60012). See top of page 70, Eq. 2, with k=1. - _N. J. A. Sloane_, Aug 31 2014
Rebekah Ann Gilbert, A Fine Rediscovery, 2014.
Amrik Singh Nimbran and Paul Levrie, Series of the form Sum {a_n*binomial(2n, n)}, Math. Student (2023) Vol. 92, Nos. 3-4, 155-173. See pp. 10, 16.

Cf. A067588, A090867, A116674, A173305.

Cf. A000009, A007360, A051424, A259936, A266591, A302569, A306200.

f:=1/(1-x^2)/product(1-x^(2*j-1),j=1..32): fser:=series(f,x=0,62): seq(coeff(fser,x,n),n=0..58); # Emeric Deutsch, Feb 22 2006

mmax = 47; CoefficientList[ Series[ (1/(1-x^2))*Product[1/(1-x^(2m+1)), {m, 0, mmax}], {x, 0, mmax}], x] (* Jean-François Alcover, Jun 21 2011 *)

# uses[EulerTransform from A166861]
def g(n): return n % 2 if n > 2 else 1
a = EulerTransform(g)
print([a(n) for n in range(48)]) # Peter Luschny, Dec 04 2020

a(n) = A036469(n) - a(n-1) = Sum_{k=0..n} (-1)^k*A036469(n-k). - Vladeta Jovovic, Sep 10 2003
a(n) = A000009(n) + a(n-2). - Vladeta Jovovic, Feb 10 2004
G.f.: 1/((1-x^2)*Product_{j>=1} (1 - x^(2*j-1))). - Emeric Deutsch, Feb 22 2006
From Vaclav Kotesovec, Aug 16 2015: (Start)
a(n) ~ (1/2) * A036469(n).
a(n) ~ 3^(1/4) * exp(Pi*sqrt(n/3)) / (4*Pi*n^(1/4)). (End)
Euler transform of the sequence [1, 1, period(1, 0)] (A266591). - Georg Fischer, Dec 04 2020

A116676 Number of odd parts in all partitions of n into distinct parts.

Original entry on oeis.org

0, 1, 0, 2, 2, 3, 4, 5, 8, 10, 14, 16, 22, 26, 34, 43, 54, 64, 80, 96, 116, 142, 170, 202, 242, 288, 340, 404, 474, 556, 652, 762, 886, 1034, 1198, 1389, 1606, 1852, 2132, 2454, 2814, 3224, 3690, 4214, 4804, 5478, 6228, 7072, 8028, 9094, 10290, 11635, 13134

Offset: 0

Emeric Deutsch, Feb 22 2006

nonn

a(n) = Sum(k*A116675(n,k), k>=0).

			a(9) = 10 because in the partitions of 9 into distinct parts, namely, [9], [81], [72], [6,3], [6,2,1], [5,4], [5,3,1] and [4,3,2], we have a total of 10 odd parts.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

Cf. A116675, A305123, A305124.

Cf. A305082, A015723, A090867, A116680, A067588.

f:=product(1+x^j,j=1..64)*sum(x^(2*j-1)/(1+x^(2*j-1)),j=1..35): fser:=series(f,x=0,60): seq(coeff(fser,x,n),n=0..56);
# second Maple program:
b:= proc(n, i) option remember; local f, g;
      if n=0 then [1, 0] elif i<1 then [0, 0]
    else f:=b(n, i-1); g:=`if`(i>n, [0, 0], b(n-i, min(n-i, i-1)));
         [f[1]+g[1], f[2]+g[2] +irem(i, 2)*g[1]]
      fi
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=0..60);  # Alois P. Heinz, Nov 21 2012

b[n_, i_] := b[n, i] = Module[{f, g}, Which [n == 0, {1, 0}, i<1 , {0, 0}, True, f = b[n, i-1]; g = If[i>n, {0, 0}, b[n-i, Min[n-i, i-1]]]; {f[[1]] + g[[1]],       f[[2]] + g[[2]] + Mod[i, 2]*g[[1]]}]]; a[n_] := b[n, n][[2]]; Table [a[n], {n, 0, 60}] (* Jean-François Alcover, May 22 2015, after Alois P. Heinz *)

G.f.: product(1+x^j, j=1..infinity)*sum(x^(2j-1)/(1+x^(2j-1)), j=1..infinity).
For n > 0, a(n) = A015723(n) - A116680(n). - Vaclav Kotesovec, May 26 2018
a(n) ~ 3^(1/4) * log(2) * exp(Pi*sqrt(n/3)) / (4*Pi*n^(1/4)). - Vaclav Kotesovec, May 26 2018

A116680 Number of even parts in all partitions of n into distinct parts.

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 4, 5, 5, 8, 11, 14, 18, 23, 29, 37, 44, 55, 69, 83, 102, 124, 148, 178, 213, 253, 300, 356, 421, 494, 582, 680, 793, 926, 1074, 1246, 1446, 1668, 1922, 2215, 2545, 2918, 3345, 3823, 4366, 4982, 5668, 6445, 7321, 8300, 9401, 10639, 12021, 13566

Offset: 0

Emeric Deutsch, Feb 22 2006

nonn

			a(9)=8 because in the partitions of 9 into distinct parts, namely, [9], [8,1], [7,2], [6,3], [6,2,1], [5,4], [5,3,1], and [4,3,2], we have a total of 8 even parts. [edited by _Rishi Advani_, Jun 07 2019]

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
D. Herden, M. R. Sepanski, J. Stanfill, C. C. Hammon, J. Henningsen, H. Ickes, J. M. Menendez, T. Poe, I. Ruiz, and E. L. Smith, Counting the parts divisible by k in all the partitions of n whose parts have multiplicity less than k, arXiv:2010.02788 [math.CO], 2020. See also Integers (2022) Vol. 22, #A49.
Runqiao Li and Andrew Y. Z. Wang, On the combinatorics of the number of even parts in all partitions with distinct parts, The Raman. J. 56 (2021) 712-727

Cf. A116679, A305121, A305122.

Cf. A305082, A015723, A090867, A067588, A116676.

m:=25; R:=PowerSeriesRing(Integers(), 3*m); [0,0] cat Coefficients(R!( (&*[1+x^j: j in [1..4*m]])*(&+[x^(2*k)/(1+x^(2*k)): k in [1..2*m]]) )); // G. C. Greubel, Jun 07 2019

f:=product(1+x^j,j=1..70)*sum(x^(2*j)/(1+x^(2*j)),j=1..40): fser:=series(f,x=0,65): seq(coeff(fser,x,n),n=0..60);
# second Maple program:
b:= proc(n, i) option remember; `if`(i*(i+1)/2 p+`if`(i::odd, 0, [0, p[1]]))(b(n-i, min(n-i, i-1)))))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..60);  # Alois P. Heinz, May 24 2022

With[{m = 25}, CoefficientList[Series[Product[1+x^j, {j,1,4*m}]* Sum[x^(2*k)/(1+x^(2*k)), {k,1,2*m}], {x,0,3*m}], x]] (* G. C. Greubel, Jun 07 2019 *)

my(m=25); my(x='x+O('x^(3*m))); concat([0, 0], Vec( prod(j=1, 4*m, 1+x^j)*sum(k=1, 2*m, x^(2*k)/(1+x^(2*k))) )) \\ G. C. Greubel, Jun 07 2019

m = 25
R = PowerSeriesRing(ZZ, 'x')
x = R.gen().O(3*m)
s = product(1+x^j for j in (1..4*m))*sum(x^(2*k)/(1+x^(2*k)) for k in (1..2*m))
[0, 0] + s.coefficients() # G. C. Greubel, Jun 07 2019

a(n) = Sum_{k >= 0} k*A116679(n,k).
G.f.: (Product_{j >= 1} (1+x^j)) * (Sum_{k >= 1} x^(2*k)/(1+x^(2*k))).
For n > 0, a(n) = A015723(n) - A116676(n). - Vaclav Kotesovec, May 26 2018
a(n) ~ 3^(1/4) * log(2) * exp(Pi*sqrt(n/3)) / (4*Pi*n^(1/4)). - Vaclav Kotesovec, May 26 2018

A305082 G.f.: Sum_{k>=1} x^k/(1-x^k) * Product_{k>=1} (1+x^k).

Original entry on oeis.org

0, 1, 3, 5, 9, 13, 20, 28, 39, 54, 71, 94, 124, 159, 201, 258, 322, 401, 499, 613, 750, 918, 1110, 1340, 1617, 1935, 2308, 2752, 3261, 3854, 4554, 5350, 6273, 7348, 8572, 9983, 11612, 13460, 15578, 18007, 20761, 23894, 27473, 31511, 36090, 41296, 47152, 53767

Offset: 0

Vaclav Kotesovec, May 25 2018

nonn

Convolution of A000005 and A000009.

Apart from initial zero this is the convolution of A341062 and A036469. - Omar E. Pol, Feb 16 2021

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000005, A000009, A006128, A015723, A209423.

Cf. A001227, A048272, A067588, A090867, A036469, A341062.

nmax = 50; CoefficientList[Series[Sum[x^k/(1-x^k), {k, 1, nmax}]*Product[1+x^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 50; CoefficientList[Series[((Log[1-x] + QPolyGamma[0, 1, x]) * QPochhammer[-1, x]) / (2*Log[x]), {x, 0, nmax}], x]

a(n) ~ 3^(1/4)*(2*gamma + log(12*n/Pi^2)) * exp(Pi*sqrt(n/3)) / (4*Pi*n^(1/4)), where gamma is the Euler-Mascheroni constant A001620.

A341494 Number of partitions of n into an even number of parts such that the set of even parts has only one element.

Original entry on oeis.org

0, 0, 0, 1, 1, 3, 1, 7, 4, 13, 6, 23, 12, 39, 20, 63, 34, 98, 53, 150, 82, 225, 124, 329, 184, 475, 267, 676, 381, 948, 539, 1317, 752, 1810, 1038, 2460, 1417, 3319, 1920, 4442, 2578, 5897, 3437, 7780, 4547, 10200, 5980, 13285, 7815, 17214, 10154, 22191, 13122

Offset: 0

Andrew Howroyd, Feb 13 2021

nonn

			The a(3) = 1 partition is: 1+2.
The a(4) = 1 partition is: 2+2.
The a(5) = 3 partitions are: 1+4, 2+3, 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, A341495, 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) - A341495(n).

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

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

Original entry on oeis.org

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

Emeric Deutsch, Dec 28 2015

nonn

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

			a(9) = 4 because we have [2,2,2,3], [3,3,3], [1,1,1,2,4], and [1,1,1,6].

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.

Column k=3 of A266477.

Cf. A015723, A090858, A090867.

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

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 *)

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

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

A305104 G.f.: Sum_{k>=1} x^(2k)/(1-x^(2k)) * Product_{k>=1} (1+x^k)/(1-x^k).

Original entry on oeis.org

0, 0, 1, 2, 6, 12, 24, 44, 79, 134, 222, 358, 566, 876, 1334, 2000, 2960, 4326, 6253, 8946, 12680, 17816, 24832, 34352, 47192, 64404, 87354, 117796, 157976, 210764, 279812, 369744, 486413, 637188, 831324, 1080420, 1398968, 1805012, 2320992, 2974728, 3800618

Offset: 0

Vaclav Kotesovec, May 25 2018

nonn

Convolution A066898 of and A000009.

Convolution A090867 of and A000041.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A015128, A066898, A090867, A305102, A305105.

nmax = 50; CoefficientList[Series[Sum[x^(2*k)/(1-x^(2*k)), {k, 1, nmax}] * Product[(1+x^k)/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ (2*gamma + log(n/Pi^2)) * exp(Pi*sqrt(n)) / (16*Pi*sqrt(n)), where gamma is the Euler-Mascheroni constant A001620.

cp's OEIS Frontend

A038348 Expansion of (1/(1-x^2))*Product_{m>=0} 1/(1-x^(2m+1)).

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A116676 Number of odd parts in all partitions of n into distinct parts.

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A116680 Number of even parts in all partitions of n into distinct parts.

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A305082 G.f.: Sum_{k>=1} x^k/(1-x^k) * Product_{k>=1} (1+x^k).

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A341494 Number of partitions of n into an even number of parts 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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A305104 G.f.: Sum_{k>=1} x^(2k)/(1-x^(2k)) * Product_{k>=1} (1+x^k)/(1-x^k).