A260890 -id:A260890 - cp's OEIS frontend

A261035 A weighted count of the number of overpartitions of n with restricted odd differences.

1, -1, -1, -1, 2, -1, 4, -5, 7, -8, 10, -15, 18, -22, 26, -37, 46, -53, 66, -84, 104, -122, 148, -183, 224, -263, 312, -379, 454, -531, 626, -750, 887, -1034, 1208, -1428, 1672, -1936, 2250, -2633, 3062, -3529, 4076, -4728, 5460, -6264, 7196, -8290, 9520, -10875, 12431, -14238

Offset: 0

Jeremy Lovejoy, Aug 07 2015

sign

The number of overpartitions of n counted with weight (-1)^(the largest part) and such that: (i) the difference between successive parts may be odd only if the larger part is overlined and (ii) if the smallest part of the overpartition is odd then it is overlined.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)
K. Bringmann, J. Dousse, J. Lovejoy, and K. Mahlburg, Overpartitions with restricted odd differences, Electron. J. Combin. 22 (2015), no.3, paper 3.17.

Cf. A260890. Equals the convolution of A141094 and A260984.

G.f.: (Product_{n >= 1} (1+q^(3*n))/(1+q^n)^3) * (1 + 2*Sum_{n >= 1} q^(n*(n+1)/2)*(1+q)^2*(1+q^2)^2*...*(1+q^(n-1))^2*(1+q^n)/((1+q^3)*(1+q^6)*...*(1+q^(3*n)))).

a(n) ~ (-1)^n * exp(2*Pi*sqrt(n)/3) / (2 * 3^(3/2) * n^(3/4)). - Vaclav Kotesovec, Jun 12 2019

A261037 The number of overpartitions of n with restricted odd differences and smallest part both odd and overlined.

Original entry on oeis.org

1, 1, 3, 4, 7, 10, 17, 23, 36, 48, 73, 96, 140, 182, 259, 334, 463, 592, 806, 1024, 1370, 1728, 2281, 2860, 3727, 4646, 5991, 7430, 9487, 11706, 14822, 18205, 22870, 27966, 34890, 42492, 52670, 63896, 78743, 95178, 116659, 140516, 171380, 205750

Offset: 1

Jeremy Lovejoy, Aug 07 2015

nonn

The number of overpartitions of n such that: (i) the difference between successive parts may be odd only if the larger part is overlined and (ii) the smallest part of the overpartition is both odd and overlined.

K. Bringmann, J. Dousse, J. Lovejoy, and K. Mahlburg, Overpartitions with restricted odd differences, Electron. J. Combin. 22 (2015), no.3, paper 3.17.

Cf. A141094, A260890, A260983, A261035.

G.f.: 1 + 3*Sum_{n >= 1} a(n)*q^n = (Product_{n >= 1} (1-q^(3*n))/((1-q^n)*(1-q^(2*n)))) * (1 + 2*Sum_{n >= 1} q^(n*(n+1)/2)*(1-q^2)*(1-q^4)*...*(1-q^(2*n-2))*(1-q^n)/((1-q^3)*(1-q^6)*...*(1-q^(3*n)))) = A260890(q)*A260983(q).

cp's OEIS Frontend

A261035 A weighted count of the number of overpartitions of n with restricted odd differences.

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