A291649 -id:A291649 - cp's OEIS frontend

A291655 Expansion of Product_{k>=1} 1/(1-x^(k^2))^(k^2).

1, 1, 1, 1, 5, 5, 5, 5, 15, 24, 24, 24, 44, 80, 80, 80, 131, 221, 266, 266, 386, 566, 746, 746, 990, 1474, 1924, 2089, 2529, 3709, 4609, 5269, 6130, 8576, 11096, 12746, 14937, 19397, 25697, 28997, 34111, 43135, 56365, 65905, 76219, 95770, 120070, 144370, 163661

Offset: 0

Vaclav Kotesovec, Aug 28 2017

nonn

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

Cf. A001156, A033461, A281790, A285047, A291649, A291696, A291720.

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

log(a(n)) ~ 5 * Pi^(1/5) * Zeta(5/2)^(2/5) * n^(3/5) / (2^(6/5) * 3^(3/5)).

A291692 Expansion of Product_{k>=1} (1+x^(k^3))^(k^3).

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 0, 8, 8, 0, 0, 0, 0, 0, 0, 28, 28, 0, 0, 0, 0, 0, 0, 56, 56, 0, 27, 27, 0, 0, 0, 70, 70, 0, 216, 216, 0, 0, 0, 56, 56, 0, 756, 756, 0, 0, 0, 28, 28, 0, 1512, 1512, 0, 351, 351, 8, 8, 0, 1890, 1890, 0, 2808, 2808, 65, 65, 0, 1512, 1512, 0

Offset: 0

Vaclav Kotesovec, Aug 30 2017

nonn

In general, if m > 0 and g.f. = Product_{k>=1} (1+x^(k^m))^(k^m), then a(n, m) ~ c * exp((2*m+1) * ((2^(1+1/m)-1) * Gamma(1/m) * Zeta(2+1/m))^(m/(2*m+1)) * n^((m+1)/(2*m+1)) / ((2*m+2)^((m+1)/(2*m+1)) * m^(3*m/(2*m+1)))) * ((2^(1+1/m)-1) * (m+1) * Gamma(1/m) * Zeta(2+1/m))^(m/(4*m+2)) / (sqrt(2*m+1) * sqrt(Pi) * 2^((3*m+2)/(4*m+2)) * m^((m-1)/(4*m+2)) * n^((3*m+1)/(4*m+2))), where c = 2^(-1/12) for m = 1 and c = 1 for m > 1.

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

Cf. A026007 (m=1), A291649 (m=2).

Cf. A033461, A279329.

nmax = 100; CoefficientList[Series[Product[(1 + x^(k^3))^(k^3), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 100; s = 1 + x; Do[s *= Sum[Binomial[k^3, j]*x^(j*k^3), {j, 0, Floor[nmax/k^3] + 1}]; s = Select[Expand[s], Exponent[#, x] <= nmax &];, {k, 2, nmax}]; Take[CoefficientList[s, x], nmax]

a(n) ~ exp(7*((2^(4/3)-1) * Gamma(1/3) * Zeta(7/3))^(3/7) * n^(4/7) / (2^(12/7) * 3^(9/7))) * ((2^(4/3)-1) * Gamma(1/3) * Zeta(7/3))^(3/14) / (2^(5/14) * 3^(1/7) * sqrt(7*Pi) * n^(5/7)).

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)).

A291696 Expansion of Product_{k>=1} (1-x^(k^2))^(k^2).

Original entry on oeis.org

1, -1, 0, 0, -4, 4, 0, 0, 6, -15, 9, 0, -4, 40, -36, 0, -15, -39, 90, -36, 64, -28, -180, 144, -96, 206, 106, -300, 148, -540, 332, 480, -232, 610, -1029, -189, 114, -86, 1880, -1068, 24, -921, -1545, 2466, -300, 2858, -1514, -3180, 976, -4121, 5590, 1995

Offset: 0

Vaclav Kotesovec, Aug 30 2017

sign

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

Cf. A073592, A291649, A291655.

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

A301518 a(n) = [x^n] Product_{k>=1} (1 + x^(k^2))^n.

Original entry on oeis.org

1, 1, 1, 1, 5, 26, 91, 246, 589, 1468, 4226, 13311, 41471, 122864, 351184, 1001876, 2920957, 8698612, 26070130, 77707056, 229959130, 679050870, 2011457295, 5986185690, 17866178695, 53343031301, 159149943668, 474683353849, 1416730630936, 4233405443596

Offset: 0

Vaclav Kotesovec, Mar 23 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A033461, A281790, A291649, A300974.

Table[SeriesCoefficient[Product[(1 + x^(k^2))^n, {k, 1, n}], {x, 0, n}], {n, 0, 30}]

a(n) ~ c * d^n / sqrt(n), where d = 3.04074590736461391963643911... and c = 0.2268848664201836146769277...

A298850 Expansion of Product_{k>=1} (1 + x^(k(k+1)/2))^(k(k+1)/2).

Original entry on oeis.org

1, 1, 0, 3, 3, 0, 9, 9, 0, 19, 29, 10, 33, 63, 30, 66, 156, 90, 110, 300, 235, 276, 561, 465, 558, 1083, 1065, 1154, 1877, 1983, 2295, 3834, 3879, 3861, 6858, 7452, 7561, 12613, 13252, 13057, 22161, 25569, 24582, 35985, 44193, 44970, 63495, 79105, 77143, 104046, 134820, 138759, 182511, 222600

Offset: 0

Ilya Gutkovskiy, Mar 09 2018

nonn

Cf. A000217, A024940, A027999, A028377, A291649, A298730.

nmax = 53; CoefficientList[Series[Product[(1 + x^(k (k + 1)/2))^(k (k + 1)/2), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 + x^A000217(k))^A000217(k).

cp's OEIS Frontend

A291655 Expansion of Product_{k>=1} 1/(1-x^(k^2))^(k^2).

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A291692 Expansion of Product_{k>=1} (1+x^(k^3))^(k^3).

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

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A291696 Expansion of Product_{k>=1} (1-x^(k^2))^(k^2).

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A301518 a(n) = [x^n] Product_{k>=1} (1 + x^(k^2))^n.

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A298850 Expansion of Product_{k>=1} (1 + x^(k*(k+1)/2))^(k*(k+1)/2).

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A298850 Expansion of Product_{k>=1} (1 + x^(k(k+1)/2))^(k(k+1)/2).