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 1-3 of 3 results.

A303904 Expansion of (1/(1 - x))*Product_{k>=1} (1 + x^(k^3)).

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 6, 6, 6, 6, 6, 6, 6, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 10, 10, 10, 10, 10, 10, 10, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13
Offset: 0

Views

Author

Ilya Gutkovskiy, May 02 2018

Keywords

Comments

Partial sums of A279329.

Links

Crossrefs

Cf. A000578, A003997, A036469, A038348, A248801, A279329, A302834.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n<0, 0,
     `if`(n=0, 1, `if`(n>i^2*(i+1)^2/4, 0, (t->
       b(t, min(t, i-1)))(n-i^3)+b(n, i-1))))
    end:
a:= proc(n) option remember; `if`(n<0, 0,
       b(n, iroot(n, 3))+a(n-1))
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, May 02 2018
  • Mathematica
    nmax = 91; CoefficientList[Series[1/(1 - x) Product[1 + x^k^3, {k, 1, Floor[nmax^(1/3) + 1]}], {x, 0, nmax}], x]

Formula

a(n) ~ exp(2^(7/4) * ((2^(1/3) - 1) * Gamma(1/3) * Zeta(4/3))^(3/4) * n^(1/4) / 3^(3/2)) * 3^(5/4) / (2^(15/8) * sqrt(Pi) * ((2^(1/3) - 1) * Gamma(1/3) * Zeta(4/3))^(3/8) * n^(1/8)). - Vaclav Kotesovec, May 04 2018

A303905 Expansion of (1/(1 - x))*Product_{k>=1} (1 + x^(k*(k+1)/2)).

Original entry on oeis.org

1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 9, 10, 10, 11, 12, 13, 15, 16, 17, 19, 20, 22, 24, 24, 26, 29, 30, 31, 34, 36, 37, 41, 44, 44, 47, 50, 52, 56, 59, 62, 65, 67, 70, 73, 75, 79, 85, 89, 91, 96, 100, 102, 108, 113, 116, 123, 129, 132, 137, 142, 147, 153, 158, 162, 169, 176, 182, 190, 196, 201
Offset: 0

Views

Author

Ilya Gutkovskiy, May 02 2018

Keywords

Comments

Partial sums of A024940.

Links

Crossrefs

Cf. A000217, A024940, A036469, A038348, A061208, A248801, A302835, A303668.

Programs

  • Mathematica
    nmax = 69; CoefficientList[Series[1/(1 - x) Product[1 + x^(k (k + 1)/2), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ exp(3 * Pi^(1/3) * ((sqrt(2) - 1) * Zeta(3/2))^(2/3) * n^(1/3) / 2^(4/3)) / (2^(1/3) * (sqrt(2) - 1)^(1/3) * sqrt(3) * Pi^(2/3) * Zeta(3/2)^(1/3) * n^(1/6)). - Vaclav Kotesovec, May 04 2018

A304633 Expansion of 2/((1 - x)*(3 + 2*x - theta_3(x))), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 2, 3, 4, 4, 4, 5, 7, 7, 7, 9, 12, 13, 13, 16, 20, 23, 23, 27, 35, 41, 42, 47, 61, 71, 75, 82, 104, 124, 134, 146, 178, 217, 237, 258, 307, 377, 419, 456, 535, 651, 739, 804, 933, 1126, 1300, 1422, 1629, 1955, 2275, 2513, 2846, 3397, 3972, 4435, 4990, 5904
Offset: 0

Views

Author

Ilya Gutkovskiy, May 15 2018

Keywords

Comments

Partial sums of A280542.

Links

Crossrefs

Cf. A000290, A001156, A006456, A010052, A248801, A280542, A302833, A303667.

Programs

  • Mathematica
    nmax = 62; CoefficientList[Series[2/((1 - x) (3 + 2 x - EllipticTheta[3, 0, x])), {x, 0, nmax}], x]
nmax = 62; CoefficientList[Series[1/((1 - x) (1 - Sum[x^k^2, {k, 2, nmax}])), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[Boole[IntegerQ[k^(1/2)] && k != 1] a[n - k], {k, 1, n}]; Accumulate[Table[a[n], {n, 0, 62}]]

Formula

G.f.: 1/((1 - x)*(1 - Sum_{k>=2} x^(k^2))).
Showing 1-3 of 3 results.

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