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 1-2 of 2 results.

A318027 Expansion of Product_{k>=1} 1/((1 - x^k)*(1 - x^(4*k))).

Original entry on oeis.org

1, 1, 2, 3, 6, 8, 13, 18, 29, 39, 57, 77, 112, 148, 205, 271, 372, 484, 647, 838, 1110, 1423, 1852, 2361, 3051, 3857, 4922, 6191, 7849, 9805, 12319, 15314, 19131, 23649, 29333, 36099, 44556, 54568, 66963, 81683, 99803, 121229, 147413, 178411, 216111, 260590, 314365, 377819, 454229
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 13 2018

Keywords

Comments

Convolution of A000041 and A035444.
Convolution of A000712 and A082303.
Convolution inverse of A107034.
Number of partitions of n if there are 2 kinds of parts that are multiples of 4.

Examples

			a(5) = 8 because we have [5], [4, 1], [4', 1], [3, 2], [3, 1, 1], [2, 2, 1], [2, 1, 1, 1] and [1, 1, 1, 1, 1].

Links

Crossrefs

Cf. A000041, A000712, A002512 (self-convolution), A002513, A035444, A082303, A100853, A107034, A318026, A318028.

Programs

  • Maple
    a:=series(mul(1/((1-x^k)*(1-x^(4*k))),k=1..55),x=0,49): seq(coeff(a,x,n),n=0..48); # Paolo P. Lava, Apr 02 2019
  • Mathematica
    nmax = 48; CoefficientList[Series[Product[1/((1 - x^k) (1 - x^(4 k))), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 48; CoefficientList[Series[1/(QPochhammer[x] QPochhammer[x^4]), {x, 0, nmax}], x]
nmax = 48; CoefficientList[Series[Exp[Sum[x^k (1 + x^k + x^(2 k) + 2*x^(3 k))/(k (1 - x^(4 k))), {k, 1, nmax}]], {x, 0, nmax}], x]
Table[Sum[PartitionsP[k] PartitionsP[n - 4 k], {k, 0, n/4}], {n, 0, 48}]

Formula

G.f.: exp(Sum_{k>=1} x^k*(1 + x^k + x^(2*k) + 2*x^(3*k))/(k*(1 - x^(4*k)))).
a(n) ~ 5^(3/4) * exp(sqrt(5*n/6)*Pi) / (2^(13/4) * 3^(3/4) * n^(5/4)). - Vaclav Kotesovec, Aug 14 2018

A318028 Expansion of Product_{k>=1} 1/((1 - x^k)*(1 - x^(5*k))).

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 12, 17, 25, 35, 51, 69, 96, 129, 175, 235, 312, 410, 539, 700, 913, 1173, 1508, 1923, 2450, 3105, 3920, 4926, 6177, 7710, 9614, 11923, 14766, 18218, 22435, 27550, 33750, 41231, 50278, 61150, 74259, 89932, 108744, 131193, 158025, 189979, 227998, 273125, 326692
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 13 2018

Keywords

Comments

Convolution of A000712 and A145466.
Convolution inverse of A030202.
Number of partitions of n if there are 2 kinds of parts that are multiples of 5.

Examples

			a(5) = 8 because we have [5], [5'], [4, 1], [3, 2], [3, 1, 1], [2, 2, 1], [2, 1, 1, 1] and [1, 1, 1, 1, 1].

Links

Crossrefs

Cf. A000712, A002513, A030202, A030205, A145466, A318026, A318027.

Programs

  • Maple
    a:=series(mul(1/((1-x^k)*(1-x^(5*k))),k=1..55),x=0,49): seq(coeff(a,x,n),n=0..48); # Paolo P. Lava, Apr 02 2019
  • Mathematica
    nmax = 48; CoefficientList[Series[Product[1/((1 - x^k) (1 - x^(5 k))), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 48; CoefficientList[Series[1/(QPochhammer[x] QPochhammer[x^5]), {x, 0, nmax}], x]
nmax = 48; CoefficientList[Series[Exp[Sum[x^k (1 + x^k + x^(2 k) + x^(3 k) + 2 x^(4 k))/(k (1 - x^(5 k))), {k, 1, nmax}]], {x, 0, nmax}], x]
Table[Sum[PartitionsP[k] PartitionsP[n - 5 k], {k, 0, n/5}], {n, 0, 48}]

Formula

G.f.: exp(Sum_{k>=1} x^k*(1 + x^k + x^(2*k) + x^(3*k) + 2*x^(4 k))/(k*(1 - x^(5*k)))).
a(n) ~ exp(2*Pi*sqrt(n/5)) / (4 * 5^(1/4) * n^(5/4)). - Vaclav Kotesovec, Aug 14 2018
Showing 1-2 of 2 results.

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