A303347 -id:A303347 - cp's OEIS frontend

A303350 Expansion of Product_{n>=1} (1 + 4*x^n)^(1/2).

1, 2, 0, 10, -10, 38, -76, 310, -960, 3190, -10672, 37262, -130170, 459690, -1639940, 5901498, -21376154, 77900710, -285457200, 1051118590, -3887169486, 14431323506, -53766825940, 200964040290, -753348868380, 2831669141514, -10670007388128

Offset: 0

Seiichi Manyama, Apr 22 2018

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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): A000009 (b=1), this sequence (b=2), A303351 (b=3).

Cf. A261568, A303347, A303352.

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

nmax = 30; CoefficientList[Series[Product[(1 + 4*x^k)^(1/2), {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+4*x^k)^(1/2)))

a(n) ~ -(-1)^n * sqrt(c) * 2^(2*n - 1) / (sqrt(Pi) * n^(3/2)), where c = Product_{k>=2} (1 + 4*(-1/4)^k) = 1.1864623436704848646891654544376222586... - Vaclav Kotesovec, Apr 22 2018

A303348 Expansion of Product_{n>=1} (1 - 9*x^n)^(1/3).

Original entry on oeis.org

1, -3, -12, -39, -246, -1578, -11487, -84054, -635781, -4893357, -38292969, -303553209, -2432865630, -19678331838, -160427322399, -1316796234933, -10872602692581, -90242886252945, -752488383572787, -6300541703215803, -52949782408528290

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.

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), A303347 (b=2), this sequence (b=3).

Cf. A303349, A303351.

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

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

a(n) ~ -c * 3^(2*n-1) / (Gamma(2/3) * n^(4/3)), where c = QPochhammer[1/9]^(1/3) = 0.95703379660353017269195329... - Vaclav Kotesovec, Apr 25 2018

cp's OEIS Frontend

A303350 Expansion of Product_{n>=1} (1 + 4*x^n)^(1/2).

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