A303361 -id:A303361 - cp's OEIS frontend

A303360 Expansion of Product_{n>=1} ((1 + 4x^n)/(1 - 4x^n))^(1/4).

1, 2, 4, 18, 34, 166, 384, 1902, 4756, 24022, 64284, 321542, 899658, 4455690, 12888944, 63185250, 187513426, 910880550, 2759413788, 13295839638, 40967821494, 195979968882, 612569599440, 2911592648458, 9213101043936, 43538337410474, 139246245625364

Offset: 0

Seiichi Manyama, Apr 22 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Expansion of Product_{n>=1} ((1 + 2^b*x^n)/(1 - 2^b*x^n))^(1/(2^b)): A015128 (b=0), A303346 (b=1), this sequence (b=2).

Cf. A303361, A303391, A067855, A303350, A303392.

seq(coeff(series(mul(((1+4*x^k)/(1-4*x^k))^(1/4), k = 1..n), x, n+1), x, n), n = 0..35); # Muniru A Asiru, Apr 22 2018

nmax = 30; CoefficientList[Series[Product[((1 + 4*x^k)/(1 - 4*x^k))^(1/4), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 22 2018 *)
nmax = 30; CoefficientList[Series[(-3*QPochhammer[-4, x] / (5*QPochhammer[4, x]))^(1/4), {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 23 2018 *)

N=66; x='x+O('x^N); Vec(prod(k=1, N, ((1+4*x^k)/(1-4*x^k))^(1/4)))

a(n) ~ c * 4^n / n^(3/4), where c = (QPochhammer[-1, 1/4] / QPochhammer[1/4])^(1/4) / Gamma(1/4) = 0.3885547372628... - Vaclav Kotesovec, Apr 23 2018

A303381 Expansion of Product_{n>=1} ((1 + (8x)^n)/(1 - (8x)^n))^(1/8).

Original entry on oeis.org

1, 2, 18, 204, 1526, 15228, 146676, 1217880, 10322982, 106429420, 886934236, 7632390312, 72137002428, 600860144728, 5351962341672, 51402944345520, 411439139563526, 3624067316629836, 33666668386023244, 279519776297893512, 2480351338204454484

Offset: 0

Seiichi Manyama, Apr 22 2018

nonn

In general, if h>=1 and g.f. = Product_{k>=1} ((1 + (h*x)^k)/(1 - (h*x)^k))^(1/h), then a(n) ~ h^n * exp(Pi*sqrt(n/h)) /(2^(3/2 + 3/(2*h)) * h^(1/4 + 1/(4*h)) * n^(3/4 + 1/(4*h))). - Vaclav Kotesovec, Apr 23 2018

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Expansion of Product_{n>=1} ((1 + (2^b*x)^n)/(1 - (2^b)*x^n))^(1/(2^b)): A015128 (b=0), A303307 (b=1), A303361 (b=2), this sequence (b=3).

Cf. A303382.

seq(coeff(series(mul(((1+(8*x)^k)/(1-(8*x)^k))^(1/8), k = 1..n), x, n+1), x, n), n = 0..25); # Muniru A Asiru, Apr 23 2018

nmax = 20; CoefficientList[Series[Product[((1 + x^k)/(1 - x^k))^(1/8), {k, 1, nmax}], {x, 0, nmax}], x] * 8^Range[0, nmax] (* Vaclav Kotesovec, Apr 23 2018 *)

N=66; x='x+O('x^N); Vec(prod(k=1, N, ((1+(8*x)^k)/(1-(8*x)^k))^(1/8)))

a(n) ~ 2^(3*n - 81/32) * exp(sqrt(n)*Pi/2^(3/2)) / n^(25/32). - Vaclav Kotesovec, Apr 23 2018

A303440 Expansion of Product_{k>=1} ((1 + (2^kx)^k)/(1 - (2^kx)^k))^(1/2^k).

Original entry on oeis.org

1, 2, 10, 148, 8502, 2114924, 2151771524, 8800410198536, 144132802083312550, 9445021284412120235340, 2475898969479166225559648172, 2596153381122039693822323043973720, 10889040933791649565507987988056678914620

Offset: 0

Seiichi Manyama, Apr 24 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..58

Cf. A303307, A303361, A303381, A303441.

N=20; x='x+O('x^N); Vec(prod(k=1, N, ((1+(2^k*x)^k)/(1-(2^k*x)^k))^(1/2^k)))

a(n) ~ 2^(n^2 - n + 1). - Vaclav Kotesovec, Apr 25 2018

A303394 Expansion of Product_{n>=1} ((1 - (4x)^n)/(1 + (4x)^n))^(1/4).

Original entry on oeis.org

1, -2, -6, -28, -26, -156, -476, 968, 11526, -16172, 139724, 791928, 1315548, 12772840, 31004424, -105335920, 1058225606, 2239259700, -3700870212, 29301955992, -4944685836, -526686535112, 1134044530040, 4057865621232, -13063873857124, -113573062924024

Offset: 0

Seiichi Manyama, Apr 23 2018

sign

Expansion of Product_{n>=1} ((1 + (2^b*x)^n)/(1 - (2^b)*x^n))^(1/(2^b)): A002448 (b=0), A303306 (b=1), this sequence (b=2), A303395 (b=3).

Cf. A303361.

N=66; x='x+O('x^N); Vec(prod(k=1, N, ((1-(4*x)^k)/(1+(4*x)^k))^(1/4)))

A303442 Expansion of Product_{k>=1} ((1 + 4^kx^k)/(1 - 4^kx^k))^(1/4^k).

Original entry on oeis.org

1, 2, 4, 18, 34, 166, 544, 2222, 5396, 29622, 101276, 411206, 1170986, 5435466, 20007472, 90854146, 253956882, 1160301990, 4412414972, 18080729238, 56012061494, 275783908498, 1010620487696, 4103148863306, 12730394683264, 58227896627114, 223877604671508

Offset: 0

Seiichi Manyama, Apr 24 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A303361, A303438, A303443.

nmax = 30; CoefficientList[Series[Exp[Sum[((-1)^j - 1) / (j*(1 - 1/(4^(j-1)*x^j))), {j, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 25 2018 *)

N=66; x='x+O('x^N); Vec(prod(k=1, N, ((1+4^k*x^k)/(1-4^k*x^k))^(1/4^k)))

cp's OEIS Frontend

A303360 Expansion of Product_{n>=1} ((1 + 4*x^n)/(1 - 4*x^n))^(1/4).

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A303381 Expansion of Product_{n>=1} ((1 + (8*x)^n)/(1 - (8*x)^n))^(1/8).

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A303440 Expansion of Product_{k>=1} ((1 + (2^k*x)^k)/(1 - (2^k*x)^k))^(1/2^k).

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A303394 Expansion of Product_{n>=1} ((1 - (4*x)^n)/(1 + (4*x)^n))^(1/4).

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A303442 Expansion of Product_{k>=1} ((1 + 4^k*x^k)/(1 - 4^k*x^k))^(1/4^k).

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A303360 Expansion of Product_{n>=1} ((1 + 4x^n)/(1 - 4x^n))^(1/4).

A303381 Expansion of Product_{n>=1} ((1 + (8x)^n)/(1 - (8x)^n))^(1/8).

A303440 Expansion of Product_{k>=1} ((1 + (2^kx)^k)/(1 - (2^kx)^k))^(1/2^k).

A303394 Expansion of Product_{n>=1} ((1 - (4x)^n)/(1 + (4x)^n))^(1/4).

A303442 Expansion of Product_{k>=1} ((1 + 4^kx^k)/(1 - 4^kx^k))^(1/4^k).