A303348 -id:A303348 - cp's OEIS frontend

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

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

A303349 Expansion of Product_{n>=1} 1/(1 - 9*x^n)^(1/3).

Original entry on oeis.org

1, 3, 21, 138, 1029, 7878, 62751, 508521, 4185885, 34819986, 292135143, 2467528563, 20958538377, 178846047741, 1532203949982, 13171424183184, 113562780734352, 981679181808261, 8505577753517235, 73846557073784937, 642328501788394527

Offset: 0

Seiichi Manyama, Apr 22 2018

nonn

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/(1 - h^2*x^k)^(1/h), then a(n) ~ h^(2*n) / (Gamma(1/h) * QPochhammer[1/h^2]^(1/h) * n^(1 - 1/h)). - Vaclav Kotesovec, Apr 22 2018

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

Expansion of Product_{n>=1} 1/(1 - b^2*x^n)^(1/b): A000041 (b=1), A067855 (b=2), this sequence (b=3).

Cf. A303348.

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

nmax = 20; CoefficientList[Series[Product[1/(1 - 9*x^k)^(1/3), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 22 2018 *)

a(n) ~ c * 3^(2*n) / n^(2/3), where c = 1 / (Gamma(1/3) * QPochhammer[1/9]^(1/3)) = 0.390040743840141117482137514... - Vaclav Kotesovec, Apr 22 2018

cp's OEIS Frontend

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

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