A281683 -id:A281683 - cp's OEIS frontend

A281781 Expansion of Product_{k>=1} (1 - x^(2k))^(2k)/(1 - x^(2k-1))^(2k-1).

1, 1, -1, 2, -1, -2, 6, -6, 3, -1, -1, 9, -18, 23, -27, 23, -1, -24, 49, -89, 121, -117, 96, -60, -18, 138, -275, 408, -525, 592, -566, 444, -181, -276, 854, -1485, 2154, -2765, 3157, -3131, 2571, -1468, -301, 2813, -5860, 9153, -12386, 15082, -16664, 16558, -14125

Offset: 0

Seiichi Manyama, Apr 14 2017

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Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A262811, A281683.

nmax = 50; CoefficientList[Series[Product[(1 - x^(2*k))^(2*k)/(1 - x^(2*k-1))^(2*k-1), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 09 2017 *)
nmax = 50; CoefficientList[Series[Product[(1 - x^(2*k))^(4*k)/(1 - x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 09 2017 *)
nmax = 50; CoefficientList[Series[Product[(1 + x^k)^(4*k)*(1 - x^k)^(3*k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 09 2017 *)

x='x+O('x^51); Vec(prod(k=1, 50, (1 - x^(2*k))^(2*k)/(1 - x^(2*k-1))^(2*k-1))) \\ Indranil Ghosh, Apr 14 2017

G.f.: exp(Sum_{k>=1} x^k/(k*(1 + x^k)^2)). - Ilya Gutkovskiy, May 28 2018

A284467 Expansion of Product_{k>=1} (1 + x^(2k-1))^(2k-1)/(1 + x^(2k))^(2k).

Original entry on oeis.org

1, 1, -2, 1, 2, -2, 0, -5, 10, 1, -15, 10, -1, 18, -39, 4, 50, -24, -14, -69, 165, -70, -83, -20, 154, 161, -550, 313, 55, 410, -960, 102, 1074, -406, -506, -1344, 3581, -1791, -833, -1833, 4995, 205, -6993, 2982, 2461, 7649, -19791, 9495, 4986, 9581, -26745, 0

Offset: 0

Seiichi Manyama, Apr 15 2017

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Robert Israel, Table of n, a(n) for n = 0..2000

Cf. A224364, A262736, A278710, A281683, A284474.

N:= 100: # to get a(0)..a(N)
P:= mul((1+x^(2*k-1))^(2*k-1)/(1+x^(2*k))^(2*k),k=1..N/2):
S:= series(P,x,N+1):
seq(coeff(S,x,j),j=0..N); # Robert Israel, Apr 16 2017

nmax = 60; CoefficientList[Series[Product[(1 + x^(2*k-1))^(2*k-1)/(1 + x^(2*k))^(2*k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 15 2017 *)

G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 + x^k)^2)). - Ilya Gutkovskiy, Jun 20 2018

cp's OEIS Frontend

A281781 Expansion of Product_{k>=1} (1 - x^(2k))^(2k)/(1 - x^(2k-1))^(2k-1).

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A284467 Expansion of Product_{k>=1} (1 + x^(2k-1))^(2k-1)/(1 + x^(2k))^(2k).

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cp's OEIS Frontend

A281781 Expansion of Product_{k>=1} (1 - x^(2*k))^(2*k)/(1 - x^(2*k-1))^(2*k-1).

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Formula

A284467 Expansion of Product_{k>=1} (1 + x^(2*k-1))^(2*k-1)/(1 + x^(2*k))^(2*k).

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A281781 Expansion of Product_{k>=1} (1 - x^(2k))^(2k)/(1 - x^(2k-1))^(2k-1).

A284467 Expansion of Product_{k>=1} (1 + x^(2k-1))^(2k-1)/(1 + x^(2k))^(2k).