A305124 -id:A305124 - 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

A305123 G.f.: Sum_{k>=1} x^(2k-1)/(1+x^(2k-1)) * Product_{k>=1} 1/(1-x^k).

Original entry on oeis.org

0, 1, 0, 3, 2, 7, 6, 15, 16, 32, 36, 62, 74, 117, 142, 214, 264, 377, 468, 648, 806, 1090, 1354, 1791, 2224, 2894, 3580, 4598, 5670, 7193, 8838, 11102, 13588, 16925, 20632, 25501, 30972, 38021, 46000, 56135, 67668, 82119, 98642, 119115, 142592, 171412, 204520

Offset: 0

Vaclav Kotesovec, May 26 2018

nonn

Conjecture: a(n) is odd iff n is a term of A067567. - Peter Bala, Jan 10 2025

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

Cf. A000041, A001227, A006128, A067567, A209423, A066898, A305121, A066897.

Cf. A116676, A305124.

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

For n > 0, a(n) = A209423(n) - A305121(n).

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

A305122 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, 4, 8, 16, 28, 47, 78, 126, 198, 306, 464, 694, 1024, 1490, 2146, 3061, 4322, 6052, 8408, 11592, 15872, 21592, 29192, 39242, 52468, 69788, 92376, 121716, 159664, 208569, 271372, 351732, 454228, 584546, 749720, 958472, 1221560, 1552210, 1966698

Offset: 0

Vaclav Kotesovec, May 26 2018

nonn

Convolution of A305121 and A000009.

The g.f. Sum_{k >= 1} x^(2*k)/(1 + x^(2*k)) * Product_{k >= 1} (1 + x ^k)/(1 - x^k) = Sum_{k >= 1} x^(2*k)/(1 + x^(2*k)) * Product_{k >= 1} (1 + x ^k)/(1 + x^k - 2*x^k) is congruent mod 2 to Sum_{k >= 1} x^(2*k)/(1 + x^(2*k)) = -G(-x^2), where G(x) is the g.f. of A112329. It follows that a(n) is odd iff n = 2*k^2 for some positive integer k. - Peter Bala, Jan 07 2025

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

Cf. A305102, A305101, A305104, A305105, A305124.

Cf. A116680, A112329, A305121.

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) = A305101(n) - A305124(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

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

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A305123 G.f.: Sum_{k>=1} x^(2*k-1)/(1+x^(2*k-1)) * Product_{k>=1} 1/(1-x^k).

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A305122 G.f.: Sum_{k>=1} x^(2*k)/(1+x^(2*k)) * 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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A305123 G.f.: Sum_{k>=1} x^(2k-1)/(1+x^(2k-1)) * Product_{k>=1} 1/(1-x^k).

A305122 G.f.: Sum_{k>=1} x^(2k)/(1+x^(2k)) * Product_{k>=1} (1+x^k)/(1-x^k).