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.

Previous Showing 11-19 of 19 results.

A226235 Expansion of q * (chi(-q) / chi(-q^3))^12 in powers of q where chi() is a Ramanujan theta function.

Original entry on oeis.org

1, -12, 66, -220, 495, -804, 1068, -1596, 3279, -6952, 12276, -17844, 23653, -34080, 57168, -98428, 154332, -215724, 285388, -395784, 600459, -931888, 1365696, -1853076, 2426189, -3277896, 4689534, -6815008, 9538632, -12664440, 16403188, -21690876, 29812932
Offset: 1

Views

Author

Michael Somos, Sep 18 2013

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = q - 12*q^2 + 66*q^3 - 220*q^4 + 495*q^5 - 804*q^6 + 1068*q^7 - 1596*q^8 + ...

Links

Crossrefs

Cf. A005259, A006353, A109389, A121665.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ q (QPochhammer[ q] QPochhammer[ q^6] / (QPochhammer[ q^2] QPochhammer[ q^3]))^12, {q, 0, n}]
nmax = 50; CoefficientList[Series[Product[((1 + x^(3*k))/(1 + x^k))^12, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 30 2017 *)
  • PARI
    {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^6 + A) / (eta(x^2 + A) * eta(x^3 + A)))^12, n))}

Formula

Expansion of q * (f(-q, -q^5) / f(-q^6))^12 in powers of q where f() is a Ramanujan theta function.
Expansion of ((c(q^2) * b(q)) / (c(q) * b(q^2)))^3 in powers of q where b() and c() are cubic AGM theta functions.
Expansion of (eta(q) * eta(q^6) / (eta(q^2) * eta(q^3)))^12 in powers of q.
Euler transform of period 6 sequence [ -12, 0, 0, 0, -12, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w)= (v - u^2) * (v - w^2) - u*w * (24*(1 + v^2) + 152*v).
G.f. A(x) satisfies f(x) = g(A(x)) where f, g are the g.f. for A006353 and A005259.
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = f(t) where q = exp(2 Pi i t).
G.f.: x * (Product_{k>0} 1 - x^k + x^(2*k))^12 where 1 - x + x^2 is the 6th cyclotomic polynomial.
Convolution inverse of A121665. Convolution 12th power of A109389.
a(n) ~ (-1)^(n+1) * exp(2*Pi*sqrt(n/3)) / (2*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Mar 30 2017
Empirical: Sum_{n>=1} a(n)/exp(2*Pi*n) = 11 + 5*sqrt(3) - sqrt(189 + 114*sqrt(3)). - Simon Plouffe, Mar 02 2021

A374064 Expansion of Product_{k>=1} 1 / (1 + x^(3*k-1)).

Original entry on oeis.org

1, 0, -1, 0, 1, -1, -1, 1, 0, -1, 1, 0, -1, 1, 0, -2, 2, 1, -3, 1, 3, -3, 0, 3, -3, -1, 4, -3, -1, 5, -3, -3, 7, -3, -5, 7, -1, -7, 8, 0, -8, 8, 1, -11, 10, 3, -14, 9, 8, -17, 8, 10, -18, 6, 14, -22, 6, 19, -24, 1, 26, -26, -3, 30, -25, -9, 37, -27, -13, 42, -26, -23, 51, -25, -31, 56
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 27 2024

Keywords

Crossrefs

Cf. A035386, A081360, A081362, A109389, A132463, A261612, A262928, A284315, A374065.

Programs

  • Mathematica
    nmax = 75; CoefficientList[Series[Product[1/(1 + x^(3 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = (1/n) Sum[DivisorSum[k, (-1)^(k/#) # &, Mod[#, 3] == 2 &] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 75}]

Formula

a(0) = 1; a(n) = -Sum_{k=1..n} A262928(k) * a(n-k).
a(n) = Sum_{k=0..n} A081362(k) * A132463(n-k).
a(n) = Sum_{k=0..n} A109389(k) * A261612(n-k).

A374065 Expansion of Product_{k>=1} 1 / (1 + x^(3*k-2)).

Original entry on oeis.org

1, -1, 1, -1, 0, 0, 0, -1, 2, -2, 1, 0, -1, 0, 2, -3, 3, -1, -1, 1, 1, -4, 5, -3, 0, 2, 0, -4, 7, -6, 1, 3, -2, -3, 9, -10, 4, 3, -5, -1, 11, -15, 10, 1, -8, 3, 10, -20, 17, -3, -10, 9, 7, -24, 26, -10, -10, 15, 2, -27, 37, -21, -8, 22, -6, -28, 49, -36, -2, 30, -19, -24, 61, -56, 10, 35
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 27 2024

Keywords

Crossrefs

Cf. A035382, A081360, A081362, A109389, A132462, A261612, A262928, A284312, A374064.

Programs

  • Mathematica
    nmax = 75; CoefficientList[Series[Product[1/(1 + x^(3 k - 2)), {k, 1, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = (1/n) Sum[DivisorSum[k, (-1)^(k/#) # &, Mod[#, 3] == 1 &] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 75}]

Formula

a(0) = 1; a(n) = -Sum_{k=1..n} A261612(k) * a(n-k).
a(n) = Sum_{k=0..n} A081362(k) * A132462(n-k).
a(n) = Sum_{k=0..n} A109389(k) * A262928(n-k).

A339404 Number of partitions of n into an even number of parts that are not multiples of 3.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 11, 13, 19, 21, 29, 35, 45, 53, 69, 80, 102, 121, 149, 176, 218, 254, 310, 365, 438, 513, 616, 716, 853, 994, 1172, 1362, 1604, 1853, 2170, 2509, 2920, 3365, 3909, 4488, 5193, 5958, 6862, 7854, 9030, 10303, 11809, 13460, 15376, 17487, 19941, 22624, 25736, 29161
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 03 2020

Keywords

Examples

			a(7) = 4 because we have [5, 2], [4, 1, 1, 1], [2, 2, 2, 1] and [2, 1, 1, 1, 1, 1].

Links

Crossrefs

Cf. A000726, A001651, A027187, A109389, A339405, A339406, A339407.

Programs

  • Maple
    b:= proc(n, i, t) option remember; `if`(n=0, t, `if`(i<1, 0,
      b(n, i-1, t)+`if`(irem(i, 3)=0, 0, b(n-i, min(n-i, i), 1-t))))
    end:
a:= n-> b(n$2, 1):
seq(a(n), n=0..60);  # Alois P. Heinz, Dec 03 2020
  • Mathematica
    nmax = 57; CoefficientList[Series[(1/2) (Product[(1 - x^(3 k))/(1 - x^k), {k, 1, nmax}] + Product[(1 + x^(3 k))/(1 + x^k), {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: (1/2) * (Product_{k>=1} (1 - x^(3*k)) / (1 - x^k) + Product_{k>=1} (1 + x^(3*k)) / (1 + x^k)).
a(n) = (A000726(n) + A109389(n)) / 2.

A276527 Expansion of Product_{k>=1} 1 / (1 + x^k - x^(2*k)).

Original entry on oeis.org

1, -1, 1, -3, 5, -8, 12, -21, 37, -59, 92, -153, 256, -409, 654, -1073, 1754, -2824, 4552, -7394, 12010, -19406, 31337, -50782, 82306, -133072, 215152, -348346, 563939, -912217, 1475604, -2388075, 3864808, -6252750, 10115987, -16369340, 26488326, -42857128
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 16 2016

Keywords

Links

Crossrefs

Cf. A003105, A162891.
Cf. A000726, A109389, A263401.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[1/Product[1+x^k-x^(2*k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ -p / (sqrt(5) * r^(n+1)), where r = -(sqrt(5)-1)/2 and p = Product_{n>1} 1/(1 + r^n - r^(2*n)) = 1.0964214808924344474065093...

A284321 Expansion of Product_{k>=0} (1 - x^(5*k+1))*(1 - x^(5*k+4)) in powers of x.

Original entry on oeis.org

1, -1, 0, 0, -1, 1, -1, 1, 0, -1, 2, -2, 1, 1, -2, 3, -3, 2, 0, -3, 5, -5, 3, 1, -5, 7, -7, 4, 1, -7, 11, -11, 6, 2, -10, 15, -15, 9, 2, -14, 22, -22, 12, 4, -20, 30, -29, 17, 4, -27, 42, -41, 23, 7, -37, 55, -54, 31, 8, -49, 76, -74, 41, 12, -66, 99, -96, 55, 14
Offset: 0

Views

Author

Seiichi Manyama, Mar 25 2017

Keywords

Links

Crossrefs

Cf. A003114, A284150, A284322, A284314, A284317.
Cf. Product_{k>=0} (1 - x^(m*k+1))*(1 - x^(m*k+m-1)): A137569 (m=3), A081362 (m=4), this sequence (m=5), A109389 (m=6).

Programs

  • Mathematica
    CoefficientList[Series[Product[(1 - x^(5k + 1)) ( 1 - x^(5k + 4)), {k, 0, 100}], {x, 0, 100}],x] (* Indranil Ghosh, Mar 25 2017 *)
  • PARI
    Vec(prod(k=0, 100, (1 - x^(5*k + 1)) * (1 - x^(5*k + 4))) + O(x^101)) \\ Indranil Ghosh, Mar 25 2017

Formula

a(n) = -(1/n)*Sum_{k=1..n} A284150(k)*a(n-k), a(0) = 1.

A339405 Number of partitions of n into an odd number of parts that are not multiples of 3.

Original entry on oeis.org

0, 1, 1, 1, 2, 3, 3, 5, 6, 8, 11, 14, 17, 23, 28, 35, 44, 55, 66, 83, 100, 122, 148, 179, 213, 259, 307, 366, 436, 518, 609, 723, 848, 997, 1169, 1369, 1593, 1864, 2163, 2513, 2914, 3376, 3894, 4503, 5182, 5965, 6854, 7869, 9008, 10325, 11794, 13470, 15363, 17509, 19911, 22654, 25713, 29177
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 03 2020

Keywords

Examples

			a(7) = 5 because we have [7], [5, 1, 1], [4, 2, 1], [2, 2, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1].

Links

Crossrefs

Cf. A000726, A001651, A027193, A109389, A339404, A339406, A339407.

Programs

  • Maple
    b:= proc(n, i, t) option remember; `if`(n=0, t, `if`(i<1, 0,
      b(n, i-1, t)+`if`(irem(i, 3)=0, 0, b(n-i, min(n-i, i), 1-t))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, Dec 03 2020
  • Mathematica
    nmax = 57; CoefficientList[Series[(1/2) (Product[(1 - x^(3 k))/(1 - x^k), {k, 1, nmax}] - Product[(1 + x^(3 k))/(1 + x^k), {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: (1/2) * (Product_{k>=1} (1 - x^(3*k)) / (1 - x^k) - Product_{k>=1} (1 + x^(3*k)) / (1 + x^k)).
a(n) = (A000726(n) - A109389(n)) / 2.

A319668 Expansion of Product_{k>=1} (1 - x^k - x^(2*k)).

Original entry on oeis.org

1, -1, -2, 0, 0, 3, 1, 3, 1, -2, 0, -3, -6, -4, 1, -8, 1, 2, 5, 5, 4, 9, 13, 7, 3, 1, 3, 7, -16, -9, -17, -13, -21, -5, -25, -33, -3, -3, -9, 22, -6, 11, 29, 29, 57, 37, 40, 31, 58, 18, 35, 40, 37, -24, -36, -34, -29, -60, -54, -98, -74, -124, -113, -156, -71, -35, -140, -46, -16, -61, -25
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 25 2018

Keywords

Crossrefs

Cf. A000010, A000358, A000726, A109389, A137569, A162891, A263401.

Programs

  • Maple
    a:=series(mul((1-x^k-x^(2*k)),k=1..100),x=0,71): seq(coeff(a,x,n),n=0..70); # Paolo P. Lava, Apr 02 2019
  • Mathematica
    nmax = 70; CoefficientList[Series[Product[(1 - x^k - x^(2 k)), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 70; CoefficientList[Series[Exp[Sum[Sum[EulerPhi[j] Log[1 - x^(j k) (1 + x^(j k))]/(j k), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, -Sum[Sum[Sum[EulerPhi[d/j] (Fibonacci[j - 1] + Fibonacci[j + 1]), {j, Divisors[d]}], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 70}]

Formula

G.f.: exp(Sum_{k>=1} Sum_{j>=1} phi(j)*log(1 - x^(j*k)*(1 + x^(j*k)))/(j*k)), where phi = Euler totient function (A000010).

A302233 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + x^(k*j))/(1 + x^j).

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -1, 1, 0, 1, -1, 0, -2, 0, 1, -1, 0, 0, 2, 0, 1, -1, 0, -1, 0, -3, 0, 1, -1, 0, -1, 2, -1, 4, 0, 1, -1, 0, -1, 1, -2, 1, -5, 0, 1, -1, 0, -1, 1, 0, 1, -1, 6, 0, 1, -1, 0, -1, 1, -1, 0, -2, 1, -8, 0, 1, -1, 0, -1, 1, -1, 2, -1, 4, 0, 10, 0, 1, -1, 0, -1, 1, -1, 1, -2, 1, -4, 0, -12, 0
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 03 2018

Keywords

Examples

			Square array begins:
1,  1,  1,  1,  1,  1,  ...
0, -1, -1, -1, -1, -1,  ...
0,  1,  0,  0,  0,  0,  ...
0, -2,  0, -1, -1, -1,  ...
0,  2,  0,  2,  1,  1,  ...
0, -3, -1, -2,  0, -1,  ...

Crossrefs

Columns k=1-10 give: A000007, A081360, A109389, A261734, A133563, A261736, A113297, A261735, A261733, A145707.
Main diagonal gives A081362.
Cf. A286653, A286656, A290307.

Programs

  • Mathematica
    Table[Function[k, SeriesCoefficient[Product[(1 + x^(k i))/(1 + x^i), {i, 1, n}], {x, 0, n}]][j - n + 1], {j, 0, 12}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[QPochhammer[-1, x^k]/QPochhammer[-1, x], {x, 0, n}]][j - n + 1], {j, 0, 12}, {n, 0, j}] // Flatten

Formula

G.f. of column k: Product_{j>=1} (1 + x^(k*j))/(1 + x^j).
For asymptotics of column k see comment from Vaclav Kotesovec in A145707.
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