A319552 -id:A319552 - cp's OEIS frontend

A341364 Expansion of (1 / theta_4(x) - 1)^3 / 8.

1, 6, 24, 77, 216, 552, 1315, 2964, 6387, 13255, 26640, 52074, 99336, 185430, 339483, 610709, 1081227, 1886484, 3247502, 5521365, 9279624, 15429149, 25397088, 41412030, 66928700, 107265576, 170556654, 269164346, 421765920, 656419080, 1015044526, 1559950185, 2383284894

Offset: 3

Ilya Gutkovskiy, Feb 10 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 3..10000

Cf. A002448, A004404, A014968, A015128, A063691, A319552, A327381, A338223, A341221, A341365, A341366, A341367, A341368, A341369, A341370.

g:= proc(n, i) option remember; `if`(n=0, 1/2, `if`(i=1, 0,
      g(n, i-1))+add(2*g(n-i*j, i-1), j=`if`(i=1, n, 1)..n/i))
    end:
b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0,
      g(n$2)), (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> b(n, 3):
seq(a(n), n=3..35);  # Alois P. Heinz, Feb 10 2021

nmax = 35; CoefficientList[Series[(1/EllipticTheta[4, 0, x] - 1)^3/8, {x, 0, nmax}], x] // Drop[#, 3] &
nmax = 35; CoefficientList[Series[(1/8) (-1 + Product[(1 + x^k)/(1 - x^k), {k, 1, nmax}])^3, {x, 0, nmax}], x] // Drop[#, 3] &

G.f.: (1/8) * (-1 + Product_{k>=1} (1 + x^k) / (1 - x^k))^3.

a(n) ~ A319552(n)/8 ~ 3*exp(Pi*sqrt(3*n)) / (512*n^(3/2)). - Vaclav Kotesovec, Feb 20 2021

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.

A319554 Expansion of 1/theta_4(q)^12 in powers of q = exp(Pi i t).

Original entry on oeis.org

1, 24, 312, 2912, 21816, 139152, 783328, 3986112, 18650424, 81251896, 332798544, 1291339296, 4776117216, 16922753616, 57683178432, 189821722688, 604884735288, 1871370360240, 5633654421720, 16535803556064, 47405095227984, 132942579098368, 365211946954656

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), A319553 (b=8), this sequence (b=12).

Cf. A002131, A002448 (theta_4(q)), A004413, A286346.

N=99; x='x+O('x^N); Vec(prod(k=1, N, ((1-x^(2*k))/(1-x^k)^2)^12))

Convolution inverse of A286346.
a(n) = (-1)^n * A004413(n).
a(0) = 1, a(n) = (24/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)^12.

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A341364 Expansion of (1 / theta_4(x) - 1)^3 / 8.

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

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A319554 Expansion of 1/theta_4(q)^12 in powers of q = exp(Pi i t).

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