A015128 -id:A015128 - cp's OEIS frontend

A303390 Expansion of Product_{k>=1} (1 + 3x^k)/(1 - 3x^k).

1, 6, 24, 96, 330, 1104, 3552, 11184, 34584, 105990, 322224, 975264, 2942016, 8857680, 26631312, 80005632, 240219114, 721036320, 2163789816, 6492625152, 19480105392, 58444390176, 175340344416, 526034008752, 1578124753152, 4734415061142, 14203316252400

Offset: 0

Vaclav Kotesovec, Apr 23 2018

nonn

Cf. A032308, A242587.

Cf. A015128, A261584, A303391.

nmax = 30; CoefficientList[Series[Product[(1+3*x^k)/(1-3*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ c * 3^n, where c = QPochhammer[-1, 1/3] / QPochhammer[1/3] = 5.5877920355220979147599292926505407983327527...

A303438 Expansion of Product_{k>=1} ((1 + 2^kx^k)/(1 - 2^kx^k))^(1/2^k).

Original entry on oeis.org

1, 2, 4, 10, 18, 38, 80, 158, 292, 630, 1260, 2470, 4922, 9706, 19392, 41010, 78466, 155494, 318764, 625670, 1238854, 2567666, 5106208, 10122522, 20022960, 40082154, 80027140, 163330106, 324201942, 643489014, 1306843568, 2592220110, 5081546084

Offset: 0

Seiichi Manyama, Apr 24 2018

nonn

a(n) / 2^n tends to 1.2036... - Vaclav Kotesovec, Apr 25 2018

Seiichi Manyama, Table of n, a(n) for n = 0..3000
Vaclav Kotesovec, Graph - The asymptotic ratio

Cf. A015128, A303346, A303360, A303382, A303439.

nmax = 30; CoefficientList[Series[Product[((1 + 2^k*x^k)/(1 - 2^k*x^k))^(1/2^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 24 2018 *)
nmax = 30; CoefficientList[Series[Exp[Sum[((-1)^j - 1) / (j*(1 - 1/(2^(j - 1)*x^j))), {j, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 25 2018 *)

my(N=66, x='x+O('x^N)); Vec(prod(k=1, N, ((1+2^k*x^k)/(1-2^k*x^k))^(1/2^k)))

G.f.: exp( Sum_{j>=1} ((-1)^j - 1) / (j*(1 - 1/(2^(j-1)*x^j))) ). - Vaclav Kotesovec, Apr 25 2018

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).

A305104 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, 6, 12, 24, 44, 79, 134, 222, 358, 566, 876, 1334, 2000, 2960, 4326, 6253, 8946, 12680, 17816, 24832, 34352, 47192, 64404, 87354, 117796, 157976, 210764, 279812, 369744, 486413, 637188, 831324, 1080420, 1398968, 1805012, 2320992, 2974728, 3800618

Offset: 0

Vaclav Kotesovec, May 25 2018

nonn

Convolution A066898 of and A000009.

Convolution A090867 of and A000041.

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

Cf. A015128, A066898, A090867, A305102, A305105.

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) ~ (2*gamma + log(n/Pi^2)) * exp(Pi*sqrt(n)) / (16*Pi*sqrt(n)), where gamma is the Euler-Mascheroni constant A001620.

A305105 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, 3, 8, 17, 34, 64, 114, 195, 325, 526, 832, 1292, 1970, 2958, 4384, 6413, 9276, 13283, 18836, 26478, 36924, 51096, 70210, 95844, 130019, 175350, 235192, 313802, 416618, 550540, 724250, 948719, 1237732, 1608508, 2082600, 2686857, 3454590, 4427144, 5655652

Offset: 0

Vaclav Kotesovec, May 25 2018

Convolution of A066897 and A000009.
Convolution of A067588 and A000041.
Let A(x) = Sum_{k >= 1} x^(2*k-1)/(1 - x^(2*k-1)) * Product_{k >= 1} (1 + x^k)/(1 - x^k). Then A(x) = Sum_{k >= 1} x^(2*k-1)/(1 - x^(2*k-1)) * Product_{k >= 1} (1 + x^k)/(1 + x^k - 2*x^k) == Sum_{k >= 1} x^(2*k-1)/(1 - x^(2*k-1)) (mod 2). It follows from the comment in A001227 by Juri-Stepan Gerasimov, dated Jul 17 2016, that a(n) is odd iff n is a square or twice a square. - Peter Bala, Jan 10 2025

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

Cf. A001227, A015128, A066897, A067588, A305102, A305104.

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) ~ (2*gamma + log(16*n/Pi^2)) * exp(Pi*sqrt(n)) / (16*Pi*sqrt(n)), where gamma is the Euler-Mascheroni constant A001620.

A316723 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j=1..k} (1+x^j)/(1-x^j).

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 2, 2, 0, 1, 2, 4, 2, 0, 1, 2, 4, 6, 2, 0, 1, 2, 4, 8, 8, 2, 0, 1, 2, 4, 8, 12, 10, 2, 0, 1, 2, 4, 8, 14, 18, 12, 2, 0, 1, 2, 4, 8, 14, 22, 26, 14, 2, 0, 1, 2, 4, 8, 14, 24, 34, 34, 16, 2, 0, 1, 2, 4, 8, 14, 24, 38, 50, 44, 18, 2, 0, 1, 2, 4, 8, 14, 24, 40, 58, 70, 56, 20, 2, 0

Offset: 0

Seiichi Manyama, Jul 11 2018

			Square array begins:
   1, 1,  1,  1,  1,  1,  1, ...
   0, 2,  2,  2,  2,  2,  2, ...
   0, 2,  4,  4,  4,  4,  4, ...
   0, 2,  6,  8,  8,  8,  8, ...
   0, 2,  8, 12, 14, 14, 14, ...
   0, 2, 10, 18, 22, 24, 24, ...
   0, 2, 12, 26, 34, 38, 40, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0-10 give: A000007, A040000, A004277, A053799, A053798, A053800, A053801, A053802, A069251, A069252, A069253.

Diagonal gives A015128.

A319552 Expansion of 1/theta_4(q)^3 in powers of q = exp(Pi i t).

Original entry on oeis.org

1, 6, 24, 80, 234, 624, 1552, 3648, 8184, 17654, 36816, 74544, 147056, 283440, 535008, 990912, 1803882, 3232224, 5707624, 9943536, 17106960, 29088352, 48922320, 81438528, 134261584, 219336630, 355242288, 570675904, 909674688, 1439394192, 2261635168, 3529838208

Offset: 0

Seiichi Manyama, Sep 22 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000

1/theta_4(q)^b: A015128 (b=1), A001934 (b=2), this sequence (b=3), A284286 (b=4), A319553 (b=8), A319554 (b=12).

Cf. A002131, A002448 (theta_4(q)), A004404, A213384.

N=99; x='x+O('x^N); Vec(prod(k=1, N, ((1-x^(2*k))/(1-x^k)^2)^3))

Convolution inverse of A213384.
a(n) = (-1)^n * A004404(n).
a(0) = 1, a(n) = (6/n)*Sum_{k=1..n} A002131(k)*a(n-k) for n > 0.
G.f.: Product_{k>=1} ((1 - x^(2k))/(1 - x^k)^2)^3.

A319553 Expansion of 1/theta_4(q)^8 in powers of q = exp(Pi i t).

Original entry on oeis.org

1, 16, 144, 960, 5264, 25056, 106944, 418176, 1520784, 5201232, 16871648, 52252992, 155341248, 445226848, 1234726272, 3323392128, 8704504976, 22234655520, 55498917840, 135595345600, 324759439584, 763505859072, 1764050361152, 4009763323008, 8975341703616, 19800832628336

Offset: 0

Seiichi Manyama, Sep 22 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000

1/theta_4(q)^b: A015128 (b=1), A001934 (b=2), A319552 (b=3), A284286 (b=4), this sequence (b=8), A319554 (b=12).

Cf. A002131, A002448 (theta_4(q)), A004409, A035016.

N=99; x='x+O('x^N); Vec(prod(k=1, N, ((1-x^(2*k))/(1-x^k)^2)^8))

Convolution inverse of A035016.
a(n) = (-1)^n * A004409(n).
a(0) = 1, a(n) = (16/n)*Sum_{k=1..n} A002131(k)*a(n-k) for n > 0.
G.f.: Product_{k>=1} ((1 - x^(2k))/(1 - x^k)^2)^8.

A330505 Expansion of e.g.f. Sum_{k>=1} arctanh(x^k).

Original entry on oeis.org

1, 2, 8, 24, 144, 960, 5760, 40320, 524160, 4354560, 43545600, 638668800, 6706022400, 99632332800, 2092278988800, 20922789888000, 376610217984000, 9247873130496000, 128047474114560000, 2919482409811968000, 77852864261652480000

Offset: 1

Ilya Gutkovskiy, Dec 16 2019

nonn

Cf. A002131, A002448, A010050, A015128, A015680, A038048, A054785, A228274, A265024, A330504.

nmax = 21; CoefficientList[Series[Sum[ArcTanh[x^k], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
nmax = 21; CoefficientList[Series[-Log[EllipticTheta[4, 0, x]]/2, {x, 0, nmax}], x] Range[0, nmax]! // Rest
Table[(n - 1)! DivisorSum[n, # &, OddQ[n/#] &], {n, 1, 21}]

E.g.f.: -log(theta_4(x)) / 2.
E.g.f.: (1/2) * Sum_{k>=1} log((1 + x^k) / (1 - x^k)).
E.g.f.: log(Product_{k>=1} ((1 + x^k) / (1 - x^k))^(1/2)).
E.g.f.: Sum_{k>=1} x^(2*k - 1) / ((2*k - 1) * (1 - x^(2*k - 1))).
exp(2 * Sum_{n>=1} a(n) * x^n / n!) = g.f. of A015128.
a(n) = (n - 1)! * Sum_{d|n, n/d odd} d.

A340659 The number of overpartitions of n having an equal number of overlined and non-overlined parts.

Original entry on oeis.org

1, 0, 1, 2, 3, 5, 7, 11, 15, 23, 31, 45, 61, 85, 114, 156, 206, 276, 363, 477, 621, 808, 1041, 1339, 1713, 2182, 2769, 3501, 4409, 5534, 6927, 8635, 10741, 13316, 16467, 20303, 24980, 30643, 37518, 45815, 55836, 67889, 82395, 99772, 120609, 145501, 175229, 210637

Offset: 0

Jeremy Lovejoy, Jan 15 2021

nonn

			a(5) = 5 counts the overpartitions [4',1], [4,1'], [3',2], [3,2'], and [2',1',1,1].

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)
B. Kim, E. Kim, and J. Lovejoy, Parity bias in partitions, European J. Combin., 89 (2020), 103159, 19 pp.

Cf. A001524, A015128, A340658.

b:= proc(n, i, c) option remember; `if`(n=0,
      `if`(c=0, 1, 0), `if`(i<1, 0, b(n, i-1, c)+add(
       add(b(n-i*j, i-1, c+j-t), t=[0, 2]), j=1..n/i)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, Jan 15 2021

b[n_, i_, c_] := b[n, i, c] = If[n==0, If[c==0, 1, 0], If[i<1, 0, b[n, i-1, c] + Sum[Sum[b[n-i*j, i-1, c+j-t], {t, {0, 2}}], {j, 1, n/i}]]];
a[n_] := b[n, n, 0];
a /@ Range[0, 60] (* Jean-François Alcover, Jan 29 2021, after Alois P. Heinz *)
nmax = 50; CoefficientList[Series[1 + Sum[x^(j*(j+1)/2 + j) / QPochhammer[x, x, j]^2, {j, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 06 2021 *)

G.f.: Sum_{n>=0} q^(n*(n+1)/2 + n)/Product_{k=1..n} (1-q^k)^2.
a(n) ~ exp(2*Pi*sqrt(n/5)) / (2^(3/2) * 5^(3/4) * phi^2 * n), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Jun 06 2021
a(n) = A143184(n) - A001524(n). - Vaclav Kotesovec, Jun 06 2021

a(0)=1 prepended by Alois P. Heinz, Jan 15 2021

cp's OEIS Frontend

A303390 Expansion of Product_{k>=1} (1 + 3*x^k)/(1 - 3*x^k).

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A303438 Expansion of Product_{k>=1} ((1 + 2^k*x^k)/(1 - 2^k*x^k))^(1/2^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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A305104 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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A305105 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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A316723 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j=1..k} (1+x^j)/(1-x^j).

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A319552 Expansion of 1/theta_4(q)^3 in powers of q = exp(Pi i t).

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A319553 Expansion of 1/theta_4(q)^8 in powers of q = exp(Pi i t).

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A330505 Expansion of e.g.f. Sum_{k>=1} arctanh(x^k).

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A303390 Expansion of Product_{k>=1} (1 + 3x^k)/(1 - 3x^k).

A303438 Expansion of Product_{k>=1} ((1 + 2^kx^k)/(1 - 2^kx^k))^(1/2^k).

A305104 G.f.: Sum_{k>=1} x^(2k)/(1-x^(2k)) * Product_{k>=1} (1+x^k)/(1-x^k).

A305105 G.f.: Sum_{k>=1} x^(2k-1)/(1-x^(2k-1)) * Product_{k>=1} (1+x^k)/(1-x^k).