cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A341365 Expansion of (1 / theta_4(x) - 1)^4 / 16.

Original entry on oeis.org

1, 8, 40, 156, 520, 1552, 4262, 10960, 26716, 62276, 139744, 303412, 640001, 1315832, 2644004, 5204044, 10052182, 19086348, 35672516, 65708116, 119409576, 214289116, 380068582, 666723748, 1157550524, 1990230968, 3390558072, 5726064688, 9590759624, 15938198484, 26289242026
Offset: 4

Views

Author

Ilya Gutkovskiy, Feb 10 2021

Keywords

Links

Crossrefs

Cf. A002448, A004405, A014968, A015128, A063730, A284286, A327382, A338223, A341222, A341364, A341366, A341367, A341368, A341369, A341370.

Programs

  • Maple
    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, 4):
seq(a(n), n=4..34);  # Alois P. Heinz, Feb 10 2021
  • Mathematica
    nmax = 34; CoefficientList[Series[(1/EllipticTheta[4, 0, x] - 1)^4/16, {x, 0, nmax}], x] // Drop[#, 4] &
nmax = 34; CoefficientList[Series[(1/16) (-1 + Product[(1 + x^k)/(1 - x^k), {k, 1, nmax}])^4, {x, 0, nmax}], x] // Drop[#, 4] &

Formula

G.f.: (1/16) * (-1 + Product_{k>=1} (1 + x^k) / (1 - x^k))^4.
a(n) ~ A284286(n)/16. - Vaclav Kotesovec, Feb 20 2021

A288515 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} ((1 + x^j)/(1 - x^j))^k.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 4, 4, 0, 1, 6, 12, 8, 0, 1, 8, 24, 32, 14, 0, 1, 10, 40, 80, 76, 24, 0, 1, 12, 60, 160, 234, 168, 40, 0, 1, 14, 84, 280, 552, 624, 352, 64, 0, 1, 16, 112, 448, 1110, 1712, 1552, 704, 100, 0, 1, 18, 144, 672, 2004, 3912, 4896, 3648, 1356, 154, 0, 1, 20, 180, 960, 3346, 7896, 12600, 13120, 8184, 2532, 232, 0
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 10 2017

Keywords

Examples

			Square array begins:
1,   1,    1,    1,     1,     1,  ...
0,   2,    4,    6,     8,    10,  ...
0,   4,   12,   24,    40,    60,  ...
0,   8,   32,   80,   160,   280,  ...
0,  14,   76,  234,   552,  1110,  ...
0,  24,  168,  624,  1712,  3913,  ...

Links

Crossrefs

Columns k=0-24 give: A000007, A015128, A001934, A004404 (alternating values), A284286, A004406-A004425 (alternating values).
Rows n=0-2 give: A000012, A005843, A046092.
Main diagonal gives A270919.
Antidiagonal sums give A299108.
Cf. A122141, A286815.

Programs

  • Julia
    # JacobiTheta4 is defined in A002448.
A288515Column(k, len) = JacobiTheta4(len, -k)
for k in 0:8 A288515Column(k, 8) |> println end # Peter Luschny, Mar 12 2018
  • Mathematica
    Table[Function[k, SeriesCoefficient[Product[((1 + x^i)/(1 - x^i))^k, {i, 1, n}], {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[1/EllipticTheta[4, 0, x]^k, {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

Formula

G.f. of column k: Product_{j>=1} ((1 + x^j)/(1 - x^j))^k.
G.f. of column k: 1/theta_4(x)^k, where theta_4() is the Jacobi theta function.
For asymptotics of column k see comment from Vaclav Kotesovec in A001934.

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

Original entry on oeis.org

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

Views

Author

Seiichi Manyama, Sep 22 2018

Keywords

Links

Crossrefs

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.

Programs

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

Formula

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

Views

Author

Seiichi Manyama, Sep 22 2018

Keywords

Links

Crossrefs

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.

Programs

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

Formula

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

Views

Author

Seiichi Manyama, Sep 22 2018

Keywords

Links

Crossrefs

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.

Programs

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

Formula

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.
Showing 1-5 of 5 results.

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