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 1-8 of 8 results.

A279226 Expansion of Product_{k>=1} (1 + x^(k^2))^2.

Original entry on oeis.org

1, 2, 1, 0, 2, 4, 2, 0, 1, 4, 5, 2, 0, 4, 8, 4, 2, 6, 7, 4, 5, 8, 6, 4, 4, 10, 15, 8, 1, 12, 24, 12, 1, 8, 19, 18, 10, 8, 16, 24, 17, 16, 23, 20, 12, 22, 34, 20, 8, 20, 42, 38, 18, 18, 42, 52, 30, 20, 34, 46, 34, 30, 46, 48, 36, 46, 72, 58, 33, 42, 71, 72, 41
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 08 2016

Keywords

Links

Crossrefs

Cf. A033461, A103265, A279225, A279227.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Product[(1 + x^(k^2))^2, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 20; poly = ConstantArray[0, nmax^2 + 1]; poly[[1]] = 1; poly[[2]] = 2; poly[[3]] = 1; Do[Do[Do[poly[[j + 1]] += poly[[j - k^2 + 1]], {j, nmax^2, k^2, -1}];, {p, 1, 2}], {k, 2, nmax}]; poly (* Vaclav Kotesovec, Dec 09 2016 *)

Formula

a(n) ~ exp(3 * Pi^(1/3) * ((sqrt(2)-1) * Zeta(3/2))^(2/3) * n^(1/3) / 2) * sqrt(2/3) * ((sqrt(2)-1) * Zeta(3/2) / Pi)^(1/3) / (4*n^(5/6)). - Vaclav Kotesovec, Dec 09 2016

A292520 Expansion of Product_{k>=1} 1/(1 + x^(k^2)).

Original entry on oeis.org

1, -1, 1, -1, 0, 0, 0, 0, 1, -2, 2, -2, 1, 0, 0, 0, 0, -1, 2, -2, 2, -1, 0, 0, 0, -1, 2, -3, 3, -2, 1, 0, 1, -2, 3, -4, 3, -2, 1, 0, 1, -2, 3, -4, 3, -2, 1, 0, 0, -2, 4, -5, 6, -4, 2, -1, 0, -2, 5, -7, 8, -6, 3, -1, 0, -1, 3, -6, 7, -6, 4, -1, 1, -1, 3, -6, 7, -8, 6, -3, 2, -4, 6, -9, 11, -9, 7, -4, 1, -3, 7
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 18 2017

Keywords

Comments

Convolution inverse of A033461.
The difference between the number of partitions of n into an even number of squares and the number of partitions of n into an odd number of squares.

Links

Crossrefs

Cf. A001156, A033461, A081362, A243148, A276516, A279225, A279226.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Product[1/(1 + x^(k^2)), {k, 1, Floor[Sqrt[nmax]] + 1}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 19 2017 *)

Formula

G.f.: Product_{k>=1} 1/(1 + x^(k^2)).
a(n) ~ (-1)^n * exp(3 * Pi^(1/3) * Zeta(3/2)^(2/3) * n^(1/3) / 2^(7/3)) * Zeta(3/2)^(1/3) / (2^(5/3) * sqrt(3) * Pi^(1/3) * n^(5/6)). - Vaclav Kotesovec, Sep 19 2017
a(n) = Sum_{k=0..n} (-1)^k * A243148(n,k). - Alois P. Heinz, Jul 25 2022

A300974 a(n) = [x^n] Product_{k>=1} 1/(1 - x^(k^2))^n.

Original entry on oeis.org

1, 1, 3, 10, 39, 151, 588, 2304, 9111, 36307, 145553, 586246, 2370264, 9614242, 39105580, 159444160, 651468967, 2666771488, 10934393619, 44899828056, 184616878289, 760010818689, 3132147583744, 12921037206764, 53351800567200, 220478125956426, 911839751015196, 3773836780169050
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 17 2018

Keywords

Comments

Number of partitions of n into squares of n kinds.

Links

Crossrefs

Cf. A000290, A001156, A008485, A023871, A240944, A279225, A285047, A300975, A301518.

Programs

  • Maple
    a:= proc(m) option remember; local b; b:= proc(n, i)
      option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      binomial(m+j-1, j)*b(n-i^2*j, i-1), j=0..n/i^2)))
      end: b(n, isqrt(n))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 17 2018
  • Mathematica
    Table[SeriesCoefficient[Product[1/(1 - x^k^2)^n, {k, 1, n}], {x, 0, n}], {n, 0, 27}]

Formula

From Vaclav Kotesovec, Mar 23 2018: (Start)
a(n) ~ c * d^n / sqrt(n), where
d = 4.216358447600641565890184638418336163396695730036... and
c = 0.26442245016754864773722176155288663999776... (End)

A302833 Expansion of (1/(1 - x))*Product_{k>=1} 1/(1 - x^(k^2)).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 12, 15, 19, 23, 27, 32, 38, 44, 50, 58, 67, 77, 87, 99, 112, 126, 140, 156, 175, 195, 216, 239, 265, 292, 320, 351, 385, 422, 460, 503, 549, 598, 648, 703, 763, 826, 892, 963, 1041, 1122, 1206, 1296, 1394, 1498, 1605, 1721, 1845, 1977, 2112, 2256, 2410, 2573
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 13 2018

Keywords

Comments

Partial sums of A001156.
Number of partitions of n into squares if there are two kinds of 1's.

Links

Crossrefs

Cf. A000070, A000290, A001156, A078134, A279225, A302835.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0 or i=1, n+1,
      b(n, i-1)+ `if`(i^2>n, 0, b(n-i^2, i)))
    end:
a:= n-> b(n, isqrt(n)):
seq(a(n), n=0..100);  # Alois P. Heinz, Apr 13 2018
  • Mathematica
    nmax = 58; CoefficientList[Series[1/(1 - x) Product[1/(1 - x^k^2), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 58; CoefficientList[Series[1/(1 - x) Sum[x^j^2/Product[(1 - x^k^2), {k, 1, j}], {j, 0, nmax}], {x, 0, nmax}], x]

Formula

G.f.: (1/(1 - x))*Sum_{j>=0} x^(j^2)/Product_{k=1..j} (1 - x^(k^2)).
From Vaclav Kotesovec, Apr 13 2018: (Start)
a(n) ~ exp(3*Pi^(1/3) * Zeta(3/2)^(2/3) * n^(1/3) / 2^(4/3)) / (2*Pi^(3/2) * sqrt(3*n)).
a(n) ~ 2^(4/3) * n^(2/3) / (Pi^(1/3) * Zeta(3/2)^(2/3)) * A001156(n). (End)

A279227 Expansion of Product_{k>=1} (1 + x^(k^2))^2/(1 - x^(k^2))^2.

Original entry on oeis.org

1, 4, 8, 12, 20, 36, 56, 76, 104, 152, 216, 284, 364, 484, 648, 828, 1028, 1300, 1664, 2076, 2532, 3108, 3848, 4700, 5640, 6776, 8200, 9848, 11660, 13796, 16424, 19452, 22776, 26612, 31240, 36572, 42440, 49092, 56968, 66044, 76040, 87236, 100280, 115244
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 08 2016

Keywords

Links

Crossrefs

Cf. A001156, A033461, A103265, A279225, A279226.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Product[(1 + x^(k^2))^2/(1 - x^(k^2))^2, {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ (4-sqrt(2)) * Zeta(3/2) * exp(3 * Pi^(1/3) * ((4-sqrt(2)) * Zeta(3/2))^(2/3) * n^(1/3) / 2^(4/3)) / (32 * sqrt(3) * Pi^2 * n^(3/2)). - Vaclav Kotesovec, Dec 29 2016

A298434 Expansion of Product_{k>=1} 1/(1 - x^(k^3))^2.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 11, 14, 17, 20, 23, 26, 29, 32, 38, 44, 50, 56, 62, 68, 74, 80, 90, 100, 110, 122, 134, 146, 158, 170, 187, 204, 221, 242, 263, 284, 305, 326, 353, 380, 407, 440, 473, 506, 539, 572, 612, 652, 692, 740, 788, 836, 887, 938, 997, 1056, 1115, 1184, 1253
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 19 2018

Keywords

Comments

Number of partitions of n into cubes of 2 kinds.
Self-convolution of A003108.

Examples

			a(8) = 11 because we have [8a], [8b], [1a, 1a, 1a, 1a, 1a, 1a, 1a, 1a], [1a, 1a, 1a, 1a, 1a, 1a, 1a, 1b], [1a, 1a, 1a, 1a, 1a, 1a, 1b, 1b], [1a, 1a, 1a, 1a, 1a, 1b, 1b, 1b], [1a, 1a, 1a, 1a, 1b, 1b, 1b, 1b], [1a, 1a, 1a, 1b, 1b, 1b, 1b, 1b], [1a, 1a, 1b, 1b, 1b, 1b, 1b, 1b], [1a, 1b, 1b, 1b, 1b, 1b, 1b, 1b] and [1b, 1b, 1b, 1b, 1b, 1b, 1b, 1b].

Links

Crossrefs

Cf. A000578, A000712, A003108, A279225.

Programs

  • Mathematica
    nmax = 60; CoefficientList[Series[Product[1/(1 - x^(k^3))^2, {k, 1, Floor[nmax^(1/3) + 1]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 08 2018 *)

Formula

G.f.: Product_{k>=1} 1/(1 - x^(k^3))^2.
a(n) ~ exp(2^(11/4) * (Gamma(1/3) * Zeta(4/3))^(3/4) * n^(1/4) / 3^(3/2)) * (Gamma(1/3) * Zeta(4/3))^(9/8) / (2^(27/8) * 3^(7/4) * Pi^(7/2) * n^(13/8)). - Vaclav Kotesovec, Apr 08 2018

A298435 Expansion of Product_{k>=1} 1/(1 - x^(k*(k+1)/2))^2.

Original entry on oeis.org

1, 2, 3, 6, 9, 12, 20, 28, 36, 52, 70, 88, 120, 156, 192, 250, 318, 386, 488, 606, 727, 900, 1101, 1308, 1590, 1916, 2257, 2706, 3225, 3768, 4465, 5270, 6117, 7178, 8399, 9686, 11274, 13094, 15020, 17352, 20017, 22846, 26230, 30080, 34175, 39010, 44500, 50346, 57184, 64914, 73156
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 19 2018

Keywords

Comments

Number of partitions of n into triangular numbers of 2 kinds.
Self-convolution of A007294.

Examples

			a(3) = 6 because we have [3a], [3b], [1a, 1a, 1a], [1a, 1a, 1b], [1a, 1b, 1b] and [1b, 1b, 1b].

Links

Crossrefs

Cf. A000217, A000712, A007294, A279225.

Programs

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

Formula

G.f.: Product_{k>=1} 1/(1 - x^(k*(k+1)/2))^2.
a(n) ~ exp(3*(Pi/2)^(1/3) * Zeta(3/2)^(2/3) * n^(1/3)) * Zeta(3/2)^(5/3) / (2^(29/6) * sqrt(3) * Pi^(5/3) * n^(13/6)). - Vaclav Kotesovec, Apr 08 2018

A329971 Expansion of 1 / (1 - 2 * Sum_{k>=1} x^(k^2)).

Original entry on oeis.org

1, 2, 4, 8, 18, 40, 88, 192, 420, 922, 2024, 4440, 9736, 21352, 46832, 102720, 225298, 494144, 1083804, 2377112, 5213736, 11435312, 25081112, 55010496, 120654744, 264632554, 580419672, 1273036832, 2792156864, 6124049048, 13431901808, 29460245120, 64615275940
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 26 2019

Keywords

Crossrefs

Cf. A000122, A004402, A006456, A025192, A032803, A240944, A279225, A279226, A280542, A320654.

Programs

  • Mathematica
    nmax = 32; CoefficientList[Series[1/(1 - 2 Sum[x^(k^2), {k, 1, Floor[Sqrt[nmax]] + 1}]), {x, 0, nmax}], x]
nmax = 32; CoefficientList[Series[1/(2 - EllipticTheta[3, 0, x]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[SquaresR[1, k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 32}]

Formula

G.f.: 1 / (2 - theta_3(x)), where theta_3() is the Jacobi theta function.
a(0) = 1; a(n) = Sum_{k=1..n} A000122(k) * a(n-k).
Showing 1-8 of 8 results.

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