A305122 -id:A305122 - cp's OEIS frontend

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

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

A305121 G.f.: Sum_{k>=1} x^(2k)/(1+x^(2k)) * Product_{k>=1} 1/(1-x^k).

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 7, 9, 14, 20, 32, 43, 63, 85, 122, 162, 221, 292, 396, 514, 680, 878, 1147, 1465, 1886, 2391, 3050, 3836, 4841, 6048, 7579, 9403, 11685, 14419, 17806, 21845, 26810, 32725, 39947, 48528, 58926, 71267, 86151, 103750, 124860, 149791, 179551

Offset: 0

Vaclav Kotesovec, May 26 2018

nonn

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

Cf. A006128, A209423, A066898, A066897, A305123.

Cf. A116680, A305122.

nmax = 60; CoefficientList[Series[Sum[x^(2*k)/(1+x^(2*k)), {k, 1, nmax}] * Product[1/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

For n > 0, a(n) = A209423(n) - A305123(n).

a(n) ~ log(2) * exp(Pi*sqrt(2*n/3)) / (2^(5/2)*Pi*sqrt(n)).

A305124 G.f.: Sum_{k>=1} x^(2k-1)/(1+x^(2k-1)) * Product_{k>=1} (1+x^k)/(1-x^k).

Original entry on oeis.org

0, 1, 1, 4, 7, 14, 24, 42, 69, 113, 178, 276, 420, 630, 930, 1360, 1963, 2804, 3969, 5568, 7746, 10700, 14672, 19986, 27060, 36423, 48754, 64928, 86038, 113478, 149012, 194842, 253737, 329172, 425452, 547952, 703343, 899858, 1147680, 1459364, 1850310, 2339432

Offset: 0

Vaclav Kotesovec, May 26 2018

nonn

Convolution of A305123 and A000009.

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

Cf. A305102, A305101, A305104, A305122, A305105.

Cf. A116676, A305123.

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

a(n) = A305101(n) - A305122(n).

a(n) ~ exp(sqrt(n)*Pi) * log(2) / (8*Pi*sqrt(n)).

A305101 G.f.: Sum_{k>=1} x^k/(1+x^k) * Product_{k>=1} (1+x^k)/(1-x^k).

Original entry on oeis.org

0, 1, 2, 6, 11, 22, 40, 70, 116, 191, 304, 474, 726, 1094, 1624, 2384, 3453, 4950, 7030, 9890, 13798, 19108, 26264, 35858, 48652, 65615, 87996, 117396, 155826, 205854, 270728, 354506, 462306, 600544, 777184, 1002180, 1287889, 1649578, 2106152, 2680924

Offset: 0

Vaclav Kotesovec, May 25 2018

nonn

Convolution of A209423 and A000009.
Convolution of A015723 and A000041.
Convolution of A048272 and A015128.
a(n) is the number of overlined parts in all overpartitions of n. - Joerg Arndt, Jun 18 2020

Cf. A006128, A015723, A209423, A305082, A305102.

Cf. A335651 and A335666.

nmax = 40; CoefficientList[Series[Sum[x^k/(1+x^k), {k, 1, nmax}] * Product[(1+x^k)/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

my(N=44, q='q+O('q^N)); Vec( prod(k=1,N, (1+q^k)/(1-q^k)) * sum(k=1,N, 1*q^k/(1+q^k)) ) \\ Joerg Arndt, Jun 18 2020

a(n) ~ exp(sqrt(n)*Pi) * log(2) / (4*Pi*sqrt(n)).

a(n) = A305122(n) + A305124(n).

cp's OEIS Frontend

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

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A305121 G.f.: Sum_{k>=1} x^(2*k)/(1+x^(2*k)) * Product_{k>=1} 1/(1-x^k).

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A305124 G.f.: Sum_{k>=1} x^(2*k-1)/(1+x^(2*k-1)) * Product_{k>=1} (1+x^k)/(1-x^k).

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A305101 G.f.: Sum_{k>=1} x^k/(1+x^k) * Product_{k>=1} (1+x^k)/(1-x^k).

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A305121 G.f.: Sum_{k>=1} x^(2k)/(1+x^(2k)) * Product_{k>=1} 1/(1-x^k).

A305124 G.f.: Sum_{k>=1} x^(2k-1)/(1+x^(2k-1)) * Product_{k>=1} (1+x^k)/(1-x^k).