A111330 -id:A111330 - cp's OEIS frontend

A284316 Expansion of Product_{k>=0} (1 - x^(4*k+3)) in powers of x.

1, 0, 0, -1, 0, 0, 0, -1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, 2, -1, 0, -1, 2, -1, 0, -1, 3, -1, 0, -2, 3, -1, 0, -3, 4, -1, 1, -4, 4, -1, 1, -5, 5, -1, 2, -7, 5, -1, 3, -8, 6, -1, 5, -10, 6, -2, 6, -12, 7, -2, 9, -14, 7, -3, 11, -16, 8, -4, 15, -19, 8, -6, 18, -21, 9

Offset: 0

Seiichi Manyama, Mar 25 2017

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

Cf. Product_{k>=0} (1 - x^(m*k+m-1)): A081362 (m=2), A284315 (m=3), this sequence (m=4), A284317 (m=5).

Cf. A050452, A147599, A284313, A111330.

CoefficientList[Series[Product[1 - x^(4k + 3), {k, 0, 100}], {x, 0, 100}], x] (* Indranil Ghosh, Mar 25 2017 *)

Vec(prod(k=0, 100, 1 - x^(4*k+3)) + O(x^101)) \\ Indranil Ghosh, Mar 25 2017

a(n) = -(1/n)*Sum_{k=1..n} A050452(k)*a(n-k), a(0) = 1.

O.g.f.: Sum_{n >= 0} (-1)^n*x^(n*(2*n+1)) / Product_{k = 1..n} ( 1 - x^(4*k) ). Cf. A284313. - Peter Bala, Nov 28 2020

A111335 Let qf(a,q) = Product_{j>=0} (1 - a*q^j); g.f. is qf(q^3,q^4)/qf(q,q^4).

Original entry on oeis.org

1, 1, 1, 0, 0, 1, 1, 0, -1, 0, 2, 1, -1, -1, 1, 2, -1, -2, 1, 3, 1, -3, -1, 3, 1, -3, -2, 4, 4, -3, -4, 3, 5, -3, -7, 2, 9, 0, -9, -1, 10, 3, -11, -5, 12, 8, -11, -10, 10, 12, -11, -15, 11, 19, -7, -21, 6, 24, -5, -28, 1, 31, 4, -33, -8, 36, 12, -38, -18, 40, 27, -40, -33, 40, 39, -40, -49, 38, 60, -34, -67, 30, 75, -25, -87, 18, 98

Offset: 0

N. J. A. Sloane, Nov 09 2005

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A111330.

# Uses EulerTransform from A358369.
a := EulerTransform(BinaryRecurrenceSequence(0, -1)):
seq(a(n), n=0..86); # Peter Luschny, Nov 17 2022

{a(n)=if(n<0, 0, polcoeff( prod(k=0,n\2, (1-x^(2*k+1))^(-(-1)^k), 1+x*O(x^n)), n))} /* Michael Somos, Nov 11 2005 */

# uses[EulerTransform from A166861]
b = BinaryRecurrenceSequence(0, -1)
a = EulerTransform(b)
print([a(n) for n in range(87)]) # Peter Luschny, Nov 17 2022

Euler transform of period 4 sequence [1, 0, -1, 0, ...]. - Michael Somos, Nov 10 2005
G.f.: Product_{k>0} (1-x^(2k-1))^((-1)^k). - Michael Somos, Nov 11 2005
G.f.: exp( Sum_{k >= 1} 1/(k*(x^k + x^(-k))) ). - Peter Bala, Sep 28 2023

cp's OEIS Frontend

A284316 Expansion of Product_{k>=0} (1 - x^(4*k+3)) in powers of x.

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