A305082 -id:A305082 - cp's OEIS frontend

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

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

A305102 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, 4, 10, 23, 46, 88, 158, 274, 459, 748, 1190, 1858, 2846, 4292, 6384, 9373, 13602, 19536, 27782, 39158, 54740, 75928, 104562, 143036, 194423, 262704, 352988, 471778, 627382, 830352, 1093994, 1435132, 1874920, 2439832, 3163020, 4085825, 5259602, 6748136

Offset: 0

Vaclav Kotesovec, May 25 2018

nonn

Convolution of A006128 and A000009.
Convolution of A305082 and A000041.
Convolution of A000005 and A015128.
a(n) is the number of non-overlined parts in all overpartitions of n. - Joerg Arndt, Jun 18 2020

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

Cf. A006128, A015723, A209423, A305082, A305101.

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(Pi*sqrt(n)) * (2*gamma + log(4*n/Pi^2)) / (8*Pi*sqrt(n)), where gamma is the Euler-Mascheroni constant A001620.

A341062 Sequence whose partial sums give A000005.

Original entry on oeis.org

1, 1, 0, 1, -1, 2, -2, 2, -1, 1, -2, 4, -4, 2, 0, 1, -3, 4, -4, 4, -2, 0, -2, 6, -5, 1, 0, 2, -4, 6, -6, 4, -2, 0, 0, 5, -7, 2, 0, 4, -6, 6, -6, 4, 0, -2, -2, 8, -7, 3, -2, 2, -4, 6, -4, 4, -4, 0, -2, 10, -10, 2, 2, 1, -3, 4, -6, 4, -2, 4, -6, 10, -10, 2, 2, 0, -2, 4, -6, 8, -5, -1, -2, 10, -8, 0, 0, 4, -6, 10

Offset: 1

Omar E. Pol, Feb 04 2021

Essentially a duplicate of A051950.
Convolved with A000041 gives A138137.
Convolved with A000027 gives the nonzero terms of A006218.
Convolved with A000070 gives the nonzero terms of A006128.
Convolved with A014153 gives the nonzero terms of A284870.
Convolved with A036469 gives the nonzero terms of A305082.
Convolved with the nonzero terms of A006218 gives A055507.
Convolved with the nonzero terms of A000217 gives the nonzero terms of A078567.

1 together with A051950.
Cf. A000005 (partial sums).
Cf. A000027, A000041, A000070, A000217, A006128, A006218, A014153, A036469, A055507, A078567, A138137, A284870, A305082, A340793.

Join[{1}, Differences[Table[DivisorSigma[0, n], {n, 1, 90}]]] (* Amiram Eldar, Feb 06 2021 *)

a(n) = A051950(n) for n > 1.

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).

A305119 G.f.: Sum_{k>=1} x^k/(1-x^k) * Product_{k>=1} 1/(1-x^k)^2.

Original entry on oeis.org

0, 1, 4, 11, 27, 58, 119, 227, 420, 744, 1287, 2160, 3561, 5739, 9113, 14224, 21924, 33327, 50126, 74531, 109802, 160211, 231875, 332821, 474313, 671072, 943411, 1317826, 1830290, 2527583, 3472446, 4746093, 6456291, 8741999, 11785768, 15822047, 21156278

Offset: 0

Vaclav Kotesovec, May 26 2018

nonn

Convolution of A006128 and A000041.

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

Cf. A000041, A000712, A006128, A305082, A305102, A305120.

nmax = 40; CoefficientList[Series[Sum[x^k/(1-x^k), {k, 1, nmax}] * Product[1/(1-x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(2*Pi*sqrt(n/3)) * (2*gamma + log(3*n/Pi^2)) / (8*3^(1/4)*Pi*n^(3/4)), where gamma is the Euler-Mascheroni constant A001620.

cp's OEIS Frontend

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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A305102 G.f.: Sum_{k>=1} x^k/(1-x^k) * Product_{k>=1} (1+x^k)/(1-x^k).

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A341062 Sequence whose partial sums give A000005.

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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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A305119 G.f.: Sum_{k>=1} x^k/(1-x^k) * Product_{k>=1} 1/(1-x^k)^2.

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