A319554 -id:A319554 - cp's OEIS frontend

A319552 Expansion of 1/theta_4(q)^3 in powers of q = exp(Pi i t).

1, 6, 24, 80, 234, 624, 1552, 3648, 8184, 17654, 36816, 74544, 147056, 283440, 535008, 990912, 1803882, 3232224, 5707624, 9943536, 17106960, 29088352, 48922320, 81438528, 134261584, 219336630, 355242288, 570675904, 909674688, 1439394192, 2261635168, 3529838208

Offset: 0

Seiichi Manyama, Sep 22 2018

nonn

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

1/theta_4(q)^b: A015128 (b=1), A001934 (b=2), this sequence (b=3), A284286 (b=4), A319553 (b=8), A319554 (b=12).

Cf. A002131, A002448 (theta_4(q)), A004404, A213384.

N=99; x='x+O('x^N); Vec(prod(k=1, N, ((1-x^(2*k))/(1-x^k)^2)^3))

Convolution inverse of A213384.
a(n) = (-1)^n * A004404(n).
a(0) = 1, a(n) = (6/n)*Sum_{k=1..n} A002131(k)*a(n-k) for n > 0.
G.f.: Product_{k>=1} ((1 - x^(2k))/(1 - x^k)^2)^3.

A319553 Expansion of 1/theta_4(q)^8 in powers of q = exp(Pi i t).

Original entry on oeis.org

1, 16, 144, 960, 5264, 25056, 106944, 418176, 1520784, 5201232, 16871648, 52252992, 155341248, 445226848, 1234726272, 3323392128, 8704504976, 22234655520, 55498917840, 135595345600, 324759439584, 763505859072, 1764050361152, 4009763323008, 8975341703616, 19800832628336

Offset: 0

Seiichi Manyama, Sep 22 2018

nonn

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

1/theta_4(q)^b: A015128 (b=1), A001934 (b=2), A319552 (b=3), A284286 (b=4), this sequence (b=8), A319554 (b=12).

Cf. A002131, A002448 (theta_4(q)), A004409, A035016.

N=99; x='x+O('x^N); Vec(prod(k=1, N, ((1-x^(2*k))/(1-x^k)^2)^8))

Convolution inverse of A035016.
a(n) = (-1)^n * A004409(n).
a(0) = 1, a(n) = (16/n)*Sum_{k=1..n} A002131(k)*a(n-k) for n > 0.
G.f.: Product_{k>=1} ((1 - x^(2k))/(1 - x^k)^2)^8.

cp's OEIS Frontend

A319552 Expansion of 1/theta_4(q)^3 in powers of q = exp(Pi i t).

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