A340647 -id:A340647 - cp's OEIS frontend

A376542 G.f.: Sum_{k>=0} x^(k^2) * Product_{j=1..k} (1 + x^(2*j))^2.

1, 1, 0, 2, 1, 1, 2, 0, 3, 1, 4, 2, 3, 3, 2, 6, 2, 7, 2, 8, 3, 10, 6, 8, 9, 8, 12, 8, 16, 6, 20, 8, 22, 10, 24, 14, 27, 20, 26, 26, 25, 34, 26, 42, 25, 51, 26, 58, 31, 66, 36, 72, 43, 76, 56, 82, 70, 82, 86, 84, 106, 87, 124, 90, 145, 95, 168, 102, 187, 115, 206

Offset: 0

Vaclav Kotesovec, Sep 28 2024

nonn

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

Cf. A216222, A306734, A333179, A340647, A369557, A376530.

nmax=100; CoefficientList[Series[Sum[x^(k^2)*Product[1+x^(2*j), {j, 1, k}]^2, {k, 0, Sqrt[nmax]}], {x, 0, nmax}], x]

a(n) ~ A369557(n) / 4.

A340623 The number of partitions of n without repeated odd parts having more even parts than odd parts.

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 3, 3, 5, 7, 8, 13, 14, 23, 23, 37, 39, 59, 63, 90, 101, 136, 156, 201, 239, 296, 355, 428, 523, 617, 754, 878, 1078, 1243, 1517, 1741, 2121, 2426, 2928, 3348, 4021, 4596, 5468, 6257, 7400, 8472, 9936, 11389, 13285, 15233, 17645, 20244, 23346

Offset: 0

Jeremy Lovejoy, Jan 13 2021

nonn

			a(7) = 3 counts the partitions [4,2,1], [3,2,2], and [2,2,2,1].

Seiichi Manyama, Table of n, a(n) for n = 0..10000
B. Kim, E. Kim, and J. Lovejoy, Parity bias in partitions, European J. Combin., 89 (2020), 103159, 19 pp.

Cf. A006950, A340621, A340622, A340647.

b:= proc(n, i, c) option remember; `if`(n=0,
      `if`(c<0, 1, 0), `if`(i<1, 0, (t-> add(b(n-i*j, i-1, c+j*
      `if`(t, 1, -1)), j=0..min(n/i, `if`(t, 1, n))))(i::odd)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, Jan 13 2021

nmax = 60; CoefficientList[Series[Product[(1 + x^(2*k - 1))/(1 - x^(2*k)), {k, 1, nmax/2}] - Sum[x^(k^2)/Product[(1 - x^(2*j))^2, {j, 1, k}], {k, 0, Sqrt[nmax]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 14 2021 *)

my(N=66, x='x+O('x^N)); concat([0, 0], Vec(prod(k=1, N, (1+x^(2*k-1))/(1-x^(2*k)))-sum(k=0, sqrt(N), x^(k^2)/prod(j=1, k, (1-x^(2*j))^2)))) \\ Seiichi Manyama, Jan 14 2021

G.f.: (Product_{k>=1} (1+q^(2*k-1))/(1-q^(2*k))) - Sum_{n>=0} q^(n^2)/Product_{k=1..n} (1-q^(2*k))^2.

A376581 G.f.: Sum_{k>=0} x^(k^2) / Product_{j=1..k} (1 - x^(2*j-1))^2.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 9, 13, 17, 22, 30, 38, 48, 62, 78, 97, 122, 151, 184, 228, 278, 335, 408, 491, 588, 707, 843, 1000, 1189, 1407, 1658, 1955, 2295, 2686, 3145, 3670, 4270, 4968, 5763, 6671, 7720, 8909, 10263, 11816, 13577, 15574, 17850, 20424, 23333, 26638, 30365

Offset: 0

Vaclav Kotesovec, Sep 29 2024

nonn

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

Cf. A053264, A306734, A340647, A376542, A376580.

nmax=100; CoefficientList[Series[Sum[x^(k^2)/Product[1-x^(2*j-1), {j, 1, k}]^2, {k, 0, Sqrt[nmax]}], {x, 0, nmax}], x]

a(n) ~ exp(Pi*sqrt(2*n/5)) / (4*5^(1/4)*sqrt(n)).

A376625 G.f.: Sum_{k>=0} x^(k(k+1)/2) / Product_{j=1..k} (1 - x^(2j))^2.

Original entry on oeis.org

1, 1, 0, 3, 0, 5, 1, 9, 2, 13, 6, 20, 12, 27, 23, 39, 40, 51, 69, 70, 108, 92, 169, 125, 252, 166, 370, 227, 527, 307, 743, 425, 1021, 586, 1393, 816, 1867, 1132, 2481, 1577, 3256, 2184, 4247, 3019, 5479, 4149, 7036, 5670, 8966, 7698, 11377, 10386, 14356, 13915, 18060

Offset: 0

Vaclav Kotesovec, Sep 30 2024

nonn

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

Cf. A000700, A072223, A179049, A340647, A376542, A376658.

nmax=80; CoefficientList[Series[Sum[x^(k*(k+1)/2)/Product[1-x^(2*j), {j, 1, k}]^2, {k, 0, Sqrt[2*nmax]}], {x, 0, nmax}], x]

G.f.: Sum_{k>=0} Product_{j=1..k} x^j/(1 - x^(2*j))^2.

a(n) ~ (r^(1/4) * sqrt(log(r)^2 + 2*polylog(2, sqrt(r))) / (2*Pi*sqrt(1 + 3*r^2))) * A376658^sqrt(n) / n, where r = A072223 = 0.52488859865640479389948613854128391569... is the smallest real root of the equation (1 - r^2)^2 = r.

cp's OEIS Frontend

A376542 G.f.: Sum_{k>=0} x^(k^2) * Product_{j=1..k} (1 + x^(2*j))^2.

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A340623 The number of partitions of n without repeated odd parts having more even parts than odd parts.

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A376581 G.f.: Sum_{k>=0} x^(k^2) / Product_{j=1..k} (1 - x^(2*j-1))^2.

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A376625 G.f.: Sum_{k>=0} x^(k(k+1)/2) / Product_{j=1..k} (1 - x^(2j))^2.

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cp's OEIS Frontend

A376542 G.f.: Sum_{k>=0} x^(k^2) * Product_{j=1..k} (1 + x^(2*j))^2.

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A340623 The number of partitions of n without repeated odd parts having more even parts than odd parts.

Views

Author

Keywords

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Programs

Maple

Mathematica

PARI

Formula

A376581 G.f.: Sum_{k>=0} x^(k^2) / Product_{j=1..k} (1 - x^(2*j-1))^2.

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A376625 G.f.: Sum_{k>=0} x^(k*(k+1)/2) / Product_{j=1..k} (1 - x^(2*j))^2.

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A376625 G.f.: Sum_{k>=0} x^(k(k+1)/2) / Product_{j=1..k} (1 - x^(2j))^2.