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.

A301504 Expansion of Product_{k>=1} (1 + x^(4*k))*(1 + x^(4*k-3)).

Original entry on oeis.org

1, 1, 0, 0, 1, 2, 1, 0, 1, 3, 2, 0, 2, 5, 4, 1, 2, 7, 7, 2, 3, 10, 11, 4, 4, 14, 17, 8, 6, 19, 25, 13, 8, 25, 36, 21, 12, 33, 50, 33, 18, 43, 69, 49, 26, 56, 93, 71, 38, 72, 124, 102, 55, 92, 163, 142, 79, 118, 212, 195, 112, 151, 273, 265, 157, 193, 350, 354, 217, 246, 444
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 22 2018

Keywords

Comments

Number of partitions of n into distinct parts congruent to 0 or 1 mod 4.

Examples

			a(9) = 3 because we have [9], [8, 1] and [5, 4].

Links

Crossrefs

Cf. A035362, A035451, A035457, A042948, A108932, A132463, A261734, A301505, A301507, A301508.

Programs

  • Mathematica
    nmax = 70; CoefficientList[Series[Product[(1 + x^(4 k)) (1 + x^(4 k - 3)), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 70; CoefficientList[Series[x^3 QPochhammer[-1, x^4] QPochhammer[-x^(-3), x^4]/(2 (1 + x) (1 - x + x^2)), {x, 0, nmax}], x]
nmax = 70; CoefficientList[Series[Product[(1 + Boole[MemberQ[{0, 1}, Mod[k, 4]]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} (1 + x^A042948(k)).
a(n) ~ exp(Pi*sqrt(n/6)) / (4*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Mar 23 2018

A301507 Expansion of Product_{k>=0} (1 + x^(4*k+1))*(1 + x^(4*k+2)).

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 2, 2, 2, 2, 2, 3, 3, 3, 4, 5, 6, 6, 6, 7, 8, 9, 10, 11, 13, 14, 14, 16, 18, 20, 23, 24, 27, 30, 31, 34, 37, 41, 46, 49, 53, 58, 62, 67, 73, 80, 88, 94, 101, 109, 117, 127, 136, 147, 161, 172, 184, 198, 211, 228, 245, 262, 284, 304, 324, 347, 370, 397, 425, 454, 488
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 22 2018

Keywords

Comments

Number of partitions of n into distinct parts congruent to 1 or 2 mod 4.

Examples

			a(11) = 3 because we have [10, 1], [9, 2] and [6, 5].

Links

Crossrefs

Cf. A035365, A042963, A070048, A108932, A169975, A301504, A301505, A301508.

Programs

  • Mathematica
    nmax = 70; CoefficientList[Series[Product[(1 + x^(4 k + 1)) (1 + x^(4 k + 2)), {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 70; CoefficientList[Series[QPochhammer[-x, x^4] QPochhammer[-x^2, x^4], {x, 0, nmax}], x]
nmax = 70; CoefficientList[Series[Product[(1 + Boole[MemberQ[{1, 2}, Mod[k, 4]]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} (1 + x^A042963(k)).
a(n) ~ exp(Pi*sqrt(n/6)) / (2^(3/2)*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Mar 23 2018
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.