A103265 -id:A103265 - cp's OEIS frontend

A279225 Expansion of Product_{k>=1} 1/(1 - x^(k^2))^2.

1, 2, 3, 4, 7, 10, 13, 16, 22, 30, 38, 46, 58, 74, 90, 106, 129, 158, 190, 222, 264, 314, 370, 426, 495, 580, 674, 772, 886, 1024, 1174, 1332, 1512, 1724, 1961, 2210, 2494, 2818, 3180, 3558, 3984, 4468, 5003, 5572, 6202, 6918, 7698, 8530, 9440, 10466, 11589

Offset: 0

Vaclav Kotesovec, Dec 08 2016

nonn

Number of partitions of n into squares of 2 kinds. - Ilya Gutkovskiy, Jan 23 2018

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

Cf. A001156, A103265, A279226, A279227.

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

a(n) ~ exp(3 * Pi^(1/3) * Zeta(3/2)^(2/3) * n^(1/3) / 2^(2/3)) * Zeta(3/2) / (8 * sqrt(3) * Pi^2 * n^(3/2)). - Vaclav Kotesovec, Dec 29 2016

A279226 Expansion of Product_{k>=1} (1 + x^(k^2))^2.

Original entry on oeis.org

1, 2, 1, 0, 2, 4, 2, 0, 1, 4, 5, 2, 0, 4, 8, 4, 2, 6, 7, 4, 5, 8, 6, 4, 4, 10, 15, 8, 1, 12, 24, 12, 1, 8, 19, 18, 10, 8, 16, 24, 17, 16, 23, 20, 12, 22, 34, 20, 8, 20, 42, 38, 18, 18, 42, 52, 30, 20, 34, 46, 34, 30, 46, 48, 36, 46, 72, 58, 33, 42, 71, 72, 41

Offset: 0

Vaclav Kotesovec, Dec 08 2016

nonn

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

Cf. A033461, A103265, A279225, A279227.

nmax = 100; CoefficientList[Series[Product[(1 + x^(k^2))^2, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 20; poly = ConstantArray[0, nmax^2 + 1]; poly[[1]] = 1; poly[[2]] = 2; poly[[3]] = 1; Do[Do[Do[poly[[j + 1]] += poly[[j - k^2 + 1]], {j, nmax^2, k^2, -1}];, {p, 1, 2}], {k, 2, nmax}]; poly (* Vaclav Kotesovec, Dec 09 2016 *)

a(n) ~ exp(3 * Pi^(1/3) * ((sqrt(2)-1) * Zeta(3/2))^(2/3) * n^(1/3) / 2) * sqrt(2/3) * ((sqrt(2)-1) * Zeta(3/2) / Pi)^(1/3) / (4*n^(5/6)). - Vaclav Kotesovec, Dec 09 2016

A280263 G.f.: Product_{k>=1} (1+x^(k^3)) / (1-x^(k^3)).

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 2, 2, 4, 6, 6, 6, 6, 6, 6, 6, 8, 10, 10, 10, 10, 10, 10, 10, 12, 14, 14, 16, 18, 18, 18, 18, 20, 22, 22, 26, 30, 30, 30, 30, 32, 34, 34, 38, 42, 42, 42, 42, 44, 46, 46, 50, 54, 54, 56, 58, 60, 62, 62, 66, 70, 70, 74, 78, 82, 86, 86, 90, 94

Offset: 0

Vaclav Kotesovec, Dec 30 2016

nonn

Convolution of A003108 and A279329.

In general, if m > 0 and g.f. = Product_{k>=1} (1 + x^(k^m)) / (1 - x^(k^m)), then a(n) ~ exp((m+1) * ((2^(1 + 1/m) - 1) * Gamma(1/m) * Zeta(1 + 1/m) / m^2)^(m/(m+1)) * (n/2)^(1/(m+1))) * ((2^(1 + 1/m) - 1) * Gamma(1/m) * Zeta(1 + 1/m))^(m/(m+1)) / (sqrt(m+1) * 2^(m/2 + (m+2)/(m+1)) * m^((3*m-1)/(2*(m+1))) * Pi^((m+1)/2) * n^((3*m+1)/(2*(m+1)))).

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

Cf. A003108, A015128, A103265, A279329.

nmax=150; CoefficientList[Series[Product[(1+x^(k^3))/(1-x^(k^3)), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(2^(7/4) * ((2^(4/3)-1) * Gamma(1/3) * Zeta(4/3))^(3/4) * n^(1/4) / 3^(3/2)) * ((2^(4/3)-1) * Gamma(1/3) * Zeta(4/3))^(3/4) / (3 * 2^(15/4) * Pi^2 * n^(5/4)).

A280366 G.f.: Product_{k>=1} (1 + x^(k(k+1)/2)) / (1 - x^(k(k+1)/2)).

Original entry on oeis.org

1, 2, 2, 4, 6, 6, 10, 14, 14, 20, 28, 30, 38, 50, 54, 66, 86, 94, 110, 138, 152, 178, 218, 238, 274, 330, 362, 412, 488, 534, 602, 710, 778, 864, 1006, 1102, 1220, 1410, 1542, 1696, 1940, 2122, 2328, 2638, 2878, 3148, 3550, 3870, 4214, 4722, 5136, 5580, 6230

Offset: 0

Vaclav Kotesovec, Jan 02 2017

nonn

Convolution of A024940 and A007294.

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

Cf. A024940, A007294, A103265.

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

a(n) ~ exp(3 * 2^(-4/3) * Pi^(1/3) * ((2*sqrt(2)-1) * Zeta(3/2))^(2/3) * n^(1/3)) * Zeta(3/2) * (2*sqrt(2)-1) / (32 * sqrt(3) * Pi * n^(3/2)).

A280276 G.f.: Product_{k>=1} (1 + x^k) / (1 - x^(k^2)).

Original entry on oeis.org

1, 2, 3, 5, 8, 12, 17, 24, 33, 46, 62, 82, 108, 141, 182, 233, 297, 375, 472, 590, 733, 907, 1117, 1369, 1671, 2034, 2465, 2978, 3586, 4304, 5152, 6149, 7319, 8689, 10293, 12162, 14340, 16871, 19806, 23207, 27139, 31678, 36909, 42932, 49851, 57794, 66897

Offset: 0

Vaclav Kotesovec, Dec 30 2016

nonn

Convolution of A000009 and A001156.

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

Cf. A000009, A001156, A087154, A103265, A280277, A369570, A369574.

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

a(n) ~ exp(Pi*sqrt(n/3) + 3^(1/4) * Zeta(3/2) * n^(1/4) / sqrt(2) - 3*Zeta(3/2)^2 / (16*Pi)) / (8*sqrt(6*Pi)*n).

A279227 Expansion of Product_{k>=1} (1 + x^(k^2))^2/(1 - x^(k^2))^2.

Original entry on oeis.org

1, 4, 8, 12, 20, 36, 56, 76, 104, 152, 216, 284, 364, 484, 648, 828, 1028, 1300, 1664, 2076, 2532, 3108, 3848, 4700, 5640, 6776, 8200, 9848, 11660, 13796, 16424, 19452, 22776, 26612, 31240, 36572, 42440, 49092, 56968, 66044, 76040, 87236, 100280, 115244

Offset: 0

Vaclav Kotesovec, Dec 08 2016

nonn

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

Cf. A001156, A033461, A103265, A279225, A279226.

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

a(n) ~ (4-sqrt(2)) * Zeta(3/2) * exp(3 * Pi^(1/3) * ((4-sqrt(2)) * Zeta(3/2))^(2/3) * n^(1/3) / 2^(4/3)) / (32 * sqrt(3) * Pi^2 * n^(3/2)). - Vaclav Kotesovec, Dec 29 2016

A306147 Expansion of e.g.f. Product_{k>=1} (1 + (exp(x)-1)^(k^2)) / (1 - (exp(x)-1)^(k^2)).

Original entry on oeis.org

1, 2, 6, 26, 198, 2042, 22566, 259226, 3249798, 47156282, 799108326, 15116875226, 305203728198, 6488119430522, 146602455461286, 3557921474016026, 92563621667899398, 2554423824661976762, 74142584637465337446, 2258422219660738881626, 72255096004023644467398

Offset: 0

Vaclav Kotesovec, Jun 23 2018

nonn

Convolution of A306082 and A306083.

Vaclav Kotesovec, Table of n, a(n) for n = 0..420

Cf. A001156, A033461, A103265, A306082, A306083.

nmax = 20; CoefficientList[Series[Product[(1 + (Exp[x] - 1)^(k^2)) / (1 - (Exp[x] - 1)^(k^2)), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!

a(n) = Sum_{k=0..n} Stirling2(n,k) * A103265(k) * k!.

a(n) ~ n! * ((2*sqrt(2) - 1) * Zeta(3/2))^(2/3) * exp(3 * (Pi/log(2))^(1/3) * ((2*sqrt(2) - 1) * Zeta(3/2))^(2/3) * n^(1/3) / 4) / (8 * sqrt(3) * Pi^(7/6) * n^(7/6) * (log(2))^(n - 1/6)).

A291666 Expansion of Product_{k>=1} ((1 + x^(k^2)) / (1 - x^(k^2)))^(k^2).

Original entry on oeis.org

1, 2, 2, 2, 10, 18, 18, 18, 50, 100, 118, 118, 206, 438, 582, 582, 806, 1606, 2344, 2506, 3122, 5322, 8202, 9498, 11130, 16844, 26110, 32272, 37018, 52274, 78018, 100098, 115986, 155026, 223190, 291674, 345132, 439518, 618734, 811790, 972846, 1204190, 1653726

Offset: 0

Vaclav Kotesovec, Aug 29 2017

nonn

Convolution of A291649 and A291655.

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

Cf. A001156, A033461, A103265, A156616, A206622, A291649, A291655, A291667, A291721.

nmax = 50; CoefficientList[Series[Product[((1+x^(k^2))/(1-x^(k^2)))^(k^2), {k, 1, nmax}], {x, 0, nmax}], x]

log(a(n)) ~ 5 * Pi^(1/5) * ((8-sqrt(2)) * Zeta(5/2))^(2/5) * n^(3/5) / (4*3^(3/5)).

A161090 Number of partitions of n into squares where every part appears at least 2 times.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 3, 2, 3, 3, 4, 3, 5, 4, 6, 5, 6, 6, 7, 6, 8, 8, 9, 9, 11, 10, 13, 11, 14, 14, 16, 15, 18, 18, 20, 19, 22, 22, 25, 24, 27, 28, 32, 29, 36, 34, 39, 38, 42, 42, 47, 45, 51, 51, 56, 55, 62, 61, 68, 66, 75, 73, 82, 79, 88, 88, 96, 93, 104, 105, 112, 113, 122, 123

Offset: 1

R. H. Hardin, Jun 02 2009

nonn

			a(12)=3 because we have 444, 441111, and 1^(12). - _Emeric Deutsch_, Jun 21 2009

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from R. H. Hardin)

Cf. A001156, A103265.

g := -1+product(1+x^(2*j^2)/(1-x^(j^2)), j = 1 .. 10): gser := series(g, x = 0, 90): seq(coeff(gser, x, n), n = 1 .. 79); # Emeric Deutsch, Jun 21 2009

nmax = 100; Rest[CoefficientList[Series[-1 + Product[(1 + x^(2*k^2)/(1-x^(k^2))), {k, 1, Sqrt[nmax] + 1}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jun 15 2025 *)
nmax = 100; Rest[CoefficientList[Series[-1 + Product[(1 + x^(3*k^2))/(1 - x^(2*k^2)), {k, 1, Sqrt[nmax/2] + 1}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jun 15 2025 *)

G.f.: -1 + Product_{j>=1} (1 + x^(2*j^2)/(1-x^(j^2))). - Emeric Deutsch, Jun 21 2009
From Vaclav Kotesovec, Jun 15 2025: (Start)
G.f.: -1 + Product_{k>=1} (1 + x^(3*k^2)) / (1 - x^(2*k^2)).
a(n) ~ ((2 - sqrt(2) + sqrt(6))*zeta(3/2))^(2/3) * exp(Pi^(1/3)*(3*(2 - sqrt(2) + sqrt(6))*zeta(3/2))^(2/3)*n^(1/3)/4) / (8 * 3^(5/6) * Pi^(7/6) * n^(7/6)). (End)

A300413 Expansion of Product_{k>=1} (1 + x^prime(k))/(1 - x^prime(k)).

Original entry on oeis.org

1, 0, 2, 2, 2, 6, 4, 10, 10, 14, 20, 22, 32, 38, 48, 60, 74, 90, 112, 134, 164, 196, 236, 282, 336, 398, 472, 554, 652, 766, 890, 1046, 1206, 1408, 1624, 1876, 2168, 2486, 2860, 3276, 3744, 4282, 4878, 5554, 6316, 7160, 8124, 9186, 10388, 11722, 13216, 14876, 16732, 18794, 21084, 23636

Offset: 0

Ilya Gutkovskiy, Mar 05 2018

nonn

Convolution of the sequences A000586 and A000607.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Index entries for related partition-counting sequences

Cf. A000040, A000586, A000607, A015128, A080054, A103265, A280263, A280366, A300414, A300415.

nmax = 55; CoefficientList[Series[Product[(1 + x^Prime[k])/(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 + x^A000040(k))/(1 - x^A000040(k)).

log(a(n)) ~ Pi*sqrt(2*n/log(n/3)) * (1/3 + 2*sqrt(log(n/3) / log(2*n/3)) / 3). - Vaclav Kotesovec, Jan 12 2021

cp's OEIS Frontend

A279225 Expansion of Product_{k>=1} 1/(1 - x^(k^2))^2.

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A279226 Expansion of Product_{k>=1} (1 + x^(k^2))^2.

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A280263 G.f.: Product_{k>=1} (1+x^(k^3)) / (1-x^(k^3)).

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A280366 G.f.: Product_{k>=1} (1 + x^(k*(k+1)/2)) / (1 - x^(k*(k+1)/2)).

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A280276 G.f.: Product_{k>=1} (1 + x^k) / (1 - x^(k^2)).

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A279227 Expansion of Product_{k>=1} (1 + x^(k^2))^2/(1 - x^(k^2))^2.

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A306147 Expansion of e.g.f. Product_{k>=1} (1 + (exp(x)-1)^(k^2)) / (1 - (exp(x)-1)^(k^2)).

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A291666 Expansion of Product_{k>=1} ((1 + x^(k^2)) / (1 - x^(k^2)))^(k^2).

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A161090 Number of partitions of n into squares where every part appears at least 2 times.

A280366 G.f.: Product_{k>=1} (1 + x^(k(k+1)/2)) / (1 - x^(k(k+1)/2)).