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 1-4 of 4 results.

A302835 Expansion of (1/(1 - x))*Product_{k>=1} 1/(1 - x^(k*(k+1)/2)).

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 13, 17, 21, 27, 34, 41, 51, 62, 73, 88, 105, 122, 144, 168, 193, 225, 260, 296, 340, 388, 438, 498, 564, 632, 713, 802, 894, 1001, 1118, 1239, 1380, 1533, 1692, 1873, 2070, 2275, 2508, 2760, 3022, 3317, 3637, 3969, 4341, 4742, 5159, 5624, 6125, 6645, 7220, 7839
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 13 2018

Keywords

Comments

Partial sums of A007294.
Number of partitions of n into triangular numbers if there are two kinds of 1's.

Links

Crossrefs

Cf. A000070, A000217, A007294, A298435, A302833.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0 or i=1, n+1,
      b(n, i-1)+(t->`if`(t>n, 0, b(n-t, i)))(i*(i+1)/2))
    end:
a:= n-> b(n, isqrt(2*n)):
seq(a(n), n=0..100);  # Alois P. Heinz, Apr 13 2018
  • Mathematica
    nmax = 55; CoefficientList[Series[1/(1 - x) Product[1/(1 - x^(k (k + 1)/2)), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 55; CoefficientList[Series[1/(1 - x) Sum[x^(j (j + 1)/2)/Product[(1 - x^(k (k + 1)/2)), {k, 1, j}], {j, 0, nmax}], {x, 0, nmax}], x]

Formula

G.f.: (1/(1 - x))*Sum_{j>=0} x^(j*(j+1)/2)/Product_{k=1..j} (1 - x^(k*(k+1)/2)).
From Vaclav Kotesovec, Apr 13 2018: (Start)
a(n) ~ exp(3*Pi^(1/3) * Zeta(3/2)^(2/3) * n^(1/3) / 2) * Zeta(3/2)^(1/3) / (2^(5/2) * sqrt(3) * Pi^(4/3) * n^(5/6)).
a(n) ~ 2 * n^(2/3) / (Pi^(1/3) * Zeta(3/2)^(2/3)) * A007294(n). (End)

A303667 Expansion of 2/((1 - x)*(3 - theta_3(x))), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 13, 18, 25, 36, 52, 74, 104, 147, 209, 297, 421, 596, 845, 1199, 1701, 2411, 3417, 4844, 6868, 9738, 13806, 19573, 27749, 39342, 55778, 79079, 112112, 158944, 225342, 319479, 452941, 642152, 910404, 1290719, 1829911, 2594344, 3678108, 5214606, 7392970, 10481335
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 28 2018

Keywords

Comments

Partial sums of A006456.

Links

Crossrefs

Cf. A000290, A006456, A010052, A302833, A303668.

Programs

  • Maple
    b:= proc(n) option remember;
      `if`(n=0, 1, add(b(n-i^2), i=1..isqrt(n)))
    end:
a:= proc(n) option remember;
      `if`(n<0, 0, b(n)+a(n-1))
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Apr 28 2018
  • Mathematica
    nmax = 45; CoefficientList[Series[2/((1 - x) (3 - EllipticTheta[3, 0, x])), {x, 0, nmax}], x]
nmax = 45; CoefficientList[Series[1/((1 - x) (1 - Sum[x^k^2, {k, 1, nmax}])), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[Boole[IntegerQ[k^(1/2)]] a[n - k], {k, 1, n}]; Accumulate[Table[a[n], {n, 0, 45}]]

Formula

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

A302834 Expansion of (1/(1 - x))*Product_{k>=1} 1/(1 - x^(k^3)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 16, 18, 20, 22, 24, 27, 30, 33, 36, 39, 42, 45, 48, 52, 56, 60, 65, 70, 75, 80, 85, 91, 97, 103, 110, 117, 124, 131, 138, 146, 154, 162, 171, 180, 189, 198, 207, 217, 227, 237, 248, 259, 270, 282, 294, 307, 320, 333, 347, 361, 375, 390, 405, 422
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 13 2018

Keywords

Comments

Partial sums of A003108.
Number of partitions of n into cubes if there are two kinds of 1's.

Links

Crossrefs

Cf. A000070, A000578, A003108, A078128, A298434, A302833.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0 or i=1, n+1,
      b(n, i-1)+ `if`(i^3>n, 0, b(n-i^3, i)))
    end:
a:= n-> b(n, iroot(n, 3)):
seq(a(n), n=0..100);  # Alois P. Heinz, Apr 13 2018
  • Mathematica
    nmax = 64; CoefficientList[Series[1/(1 - x) Product[1/(1 - x^k^3), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 64; CoefficientList[Series[1/(1 - x) Sum[x^j^3/Product[(1 - x^k^3), {k, 1, j}], {j, 0, nmax}], {x, 0, nmax}], x]

Formula

G.f.: (1/(1 - x))*Sum_{j>=0} x^(j^3)/Product_{k=1..j} (1 - x^(k^3)).
From Vaclav Kotesovec, Apr 13 2018: (Start)
a(n) ~ sqrt(3) * exp(4*(Gamma(1/3)*Zeta(4/3))^(3/4) * n^(1/4) / 3^(3/2)) / (8 * Pi^2 * sqrt(n)).
a(n) ~ 3^(3/2) * n^(3/4) / (Gamma(1/3)*Zeta(4/3))^(3/4) * A003108(n). (End)

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 1-4 of 4 results.

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