A303350 -id:A303350 - 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

A303391 Expansion of Product_{k>=1} (1 + 4x^k)/(1 - 4x^k).

Original entry on oeis.org

1, 8, 40, 200, 872, 3720, 15400, 62920, 254440, 1024648, 4112680, 16483400, 66000360, 264150920, 1056903080, 4228272200, 16914393832, 67660396040, 270647139240, 1082600410440, 4330424811880, 17321748357640, 69287088965800, 277148557003720, 1108594618342760

Offset: 0

Vaclav Kotesovec, Apr 23 2018

nonn

Robert Israel, Table of n, a(n) for n = 0..1600

Cf. A261568, A246936, A067855, A303350.

Cf. A015128, A261584, A303390.

N:= 50: # for a(0)..a(N)
G:= mul((1+4*x^k)/(1-4*x^k),k=1..N):
S:= series(G,x,N+1):
seq(coeff(S,x,j),j=0..N); # Robert Israel, Feb 13 2019

nmax = 25; CoefficientList[Series[Product[(1+4*x^k)/(1-4*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ c * 4^n, where c = QPochhammer[-1, 1/4] / QPochhammer[1/4] = 3.9385207073365388638943873939345313401323799...

A303347 Expansion of Product_{n>=1} (1 - 4*x^n)^(1/2).

Original entry on oeis.org

1, -2, -4, -2, -6, -6, -56, -158, -612, -2070, -7228, -25238, -89646, -319466, -1150168, -4164978, -15177718, -55592614, -204617788, -756314982, -2806456898, -10450497682, -39040372248, -146273912858, -549533738952, -2069680656234, -7812908945556

Offset: 0

Seiichi Manyama, Apr 22 2018

sign

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -1/2, g(n) = 4.

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

Expansion of Product_{n>=1} (1 - b^2*x^n)^(1/b): A010815 (b=1), this sequence (b=2), A303348 (b=3).

Cf. A067855, A303350.

seq(coeff(series(mul((1-4*x^k)^(1/2), k = 1..n), x, n+1), x, n), n=0..40); # Muniru A Asiru, Apr 22 2018

N=66; x='x+O('x^N); Vec(prod(k=1, N, (1-4*x^k)^(1/2)))

a(n) ~ -c * 2^(2*n-1) / (sqrt(Pi) * n^(3/2)), where c = QPochhammer[1/4]^(1/2) = 0.8297816201389011939293261374110190... - Vaclav Kotesovec, Apr 25 2018

A303352 Expansion of Product_{n>=1} 1/(1 + 4*x^n)^(1/2).

Original entry on oeis.org

1, -2, 4, -18, 66, -230, 832, -3118, 11764, -44374, 168476, -643974, 2470506, -9503946, 36666736, -141824034, 549717490, -2134650662, 8303024092, -32343942934, 126161860886, -492703658930, 1926278860624, -7538530620746, 29529208903872, -115766389203370

Offset: 0

Seiichi Manyama, Apr 22 2018

sign

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = 1/2, g(n) = -4.

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

Expansion of Product_{n>=1} 1/(1 + b^2*x^n)^(1/b): A081362 (b=1), this sequence (b=2), A303353 (b=3).

Cf. A067855, A303350.

seq(coeff(series(mul(1/(1+4*x^k)^(1/2), k = 1..n), x, n+1), x, n), n=0..40); # Muniru A Asiru, Apr 22 2018

nmax = 30; CoefficientList[Series[Product[1/(1 + 4*x^k)^(1/2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 25 2018 *)

a(n) ~ c * (-4)^n / sqrt(Pi*n), where c = 1 / QPochhammer[-1/4]^(1/2) = 0.91806413264267465793225216525758518... - Vaclav Kotesovec, Apr 25 2018

A303351 Expansion of Product_{n>=1} (1 + 9*x^n)^(1/3).

Original entry on oeis.org

1, 3, -6, 57, -294, 1884, -13011, 95178, -712293, 5448495, -42444375, 335392941, -2681006280, 21639853488, -176113016241, 1443450932445, -11903668996713, 98695838478585, -822212761531101, 6878755556938029, -57767592614370576, 486792969548157129

Offset: 0

Seiichi Manyama, Apr 22 2018

sign

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -1/3, g(n) = -9.

In general, if h > 1 and g.f. = Product_{k>=1} (1 + h^2*x^k)^(1/h), then a(n) ~ -(-1)^n * c^(1/h) * h^(2*n-1) / (Gamma(1 - 1/h) * n^(1 + 1/h)), where c = Product_{k>=2} (1 + (-1)^k / h^(2*k-2)). - Vaclav Kotesovec, Apr 22 2018

Expansion of Product_{n>=1} (1 + b^2*x^n)^(1/b): A000009 (b=1), A303350 (b=2), this sequence (b=3).

seq(coeff(series(mul((1+9*x^k)^(1/3), k = 1..n), x, n+1), x, n), n = 0..25); # Muniru A Asiru, Apr 22 2018

nmax = 30; CoefficientList[Series[Product[(1 + 9*x^k)^(1/3), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 22 2018 *)

N=66; x='x+O('x^N); Vec(prod(k=1, N, (1+9*x^k)^(1/3)))

a(n) ~ -(-1)^n * c^(1/3) * 3^(2*n-1) / (Gamma(2/3) * n^(4/3)), where c = Product_{k>=2} (1 + 9*(-1/9)^k) = 1.09874828793226302381837574278380702... - Vaclav Kotesovec, Apr 22 2018

A303392 Expansion of Product_{k>=1} ((1 + 4x^k) / (1 - 4x^k))^(1/2).

Original entry on oeis.org

1, 4, 12, 52, 156, 612, 2028, 7892, 27324, 107396, 384844, 1520436, 5566876, 22069796, 81990252, 325707348, 1222582268, 4862950020, 18395472460, 73233825524, 278700724764, 1110232691108, 4245596648876, 16920914168148, 64963831455996, 259012955299396

Offset: 0

Vaclav Kotesovec, Apr 23 2018

nonn

Cf. A261568, A246936, A067855, A303350, A303391, A303360.

nmax = 30; CoefficientList[Series[Product[((1+4*x^k)/(1-4*x^k))^(1/2), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 30; CoefficientList[Series[(-3*QPochhammer[-4, x] / (5*QPochhammer[4, x]))^(1/2), {x, 0, nmax}], x]

a(n) ~ sqrt(c) * 4^n / sqrt(Pi*n), where c = QPochhammer[-1, 1/4]/QPochhammer[1/4] = 3.9385207073365388638943873939345313401323799...

cp's OEIS Frontend

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

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A303391 Expansion of Product_{k>=1} (1 + 4*x^k)/(1 - 4*x^k).

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A303347 Expansion of Product_{n>=1} (1 - 4*x^n)^(1/2).

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A303352 Expansion of Product_{n>=1} 1/(1 + 4*x^n)^(1/2).

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A303351 Expansion of Product_{n>=1} (1 + 9*x^n)^(1/3).

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A303392 Expansion of Product_{k>=1} ((1 + 4*x^k) / (1 - 4*x^k))^(1/2).

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

A303391 Expansion of Product_{k>=1} (1 + 4x^k)/(1 - 4x^k).

A303392 Expansion of Product_{k>=1} ((1 + 4x^k) / (1 - 4x^k))^(1/2).