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.

A277210 Expansion of Product_{k>=1} 1/(1 - x^(3*k+1)).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 4, 3, 3, 4, 5, 4, 6, 6, 7, 7, 9, 8, 11, 11, 12, 13, 16, 15, 18, 20, 22, 22, 27, 27, 31, 33, 37, 38, 45, 46, 51, 55, 62, 63, 72, 76, 84, 89, 99, 103, 116, 122, 133, 142, 158, 164, 181, 193, 210, 222, 245, 257, 281, 299, 324, 343, 376, 396, 429, 457, 495, 522, 568, 601, 649, 689
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 05 2016

Keywords

Comments

Number of partitions of n into parts larger than 1 and congruent to 1 mod 3.
More generally, the ordinary generating function for the number of partitions of n into parts larger than 1 and congruent to 1 mod m (for m>0) is Product_{k>=1} 1/(1 - x^(m*k+1)).

Examples

			a(14) = 2, because we have [10, 4] and [7, 7].

Links

Crossrefs

Cf. A016777, A035382, A087897, A117957.

Programs

  • Mathematica
    CoefficientList[Series[(1 - x)/QPochhammer[x, x^3], {x, 0, 85}], x]

Formula

G.f.: Product_{k>=1} 1/(1 - x^(3*k+1)).
a(n) ~ Pi^(1/3) * Gamma(1/3) * exp(sqrt(2*n)*Pi/3) / (2^(13/6)*3^(3/2)*n^(7/6)). - Vaclav Kotesovec, Oct 06 2016

A277264 Expansion of Product_{k>=1} 1/(1 - x^(5*k+1)).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 2, 1, 1, 0, 1, 2, 2, 1, 1, 1, 3, 3, 2, 1, 2, 3, 4, 3, 2, 2, 5, 5, 5, 3, 3, 5, 8, 6, 5, 4, 7, 9, 10, 7, 6, 8, 12, 12, 11, 8, 11, 15, 17, 14, 13, 13, 19, 21, 20, 16, 19, 23, 28, 26, 23, 23, 31, 34, 35, 30, 31, 37, 46, 44, 41, 39, 48, 55, 59, 52, 52, 59, 71, 73, 71, 65, 75, 87, 94
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 07 2016

Keywords

Comments

Number of partitions of n into parts larger than 1 and congruent to 1 mod 5.

Examples

			a(22) = 2, because we have [16, 6] and [11, 11].

Links

Crossrefs

Cf. A016861, A087897, A109697 (partial sums), A117957, A277210.

Programs

  • Mathematica
    CoefficientList[Series[(1 - x)/QPochhammer[x, x^5], {x, 0, 100}], x]

Formula

G.f.: Product_{k>=1} 1/(1 - x^(5*k+1)).
a(n) ~ Pi^(1/5) * Gamma(1/5) * exp(Pi*sqrt(2*n/15)) / (2^(21/10) * 3^(3/5) * 5^(9/10) * n^(11/10)). - Vaclav Kotesovec, Oct 09 2016

A277349 Expansion of Product_{k>=1} 1/(1 - x^(6*k+1)).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 2, 1, 1, 0, 0, 1, 2, 2, 1, 1, 0, 1, 3, 3, 2, 1, 1, 1, 3, 4, 3, 2, 1, 2, 4, 5, 5, 3, 2, 2, 5, 7, 6, 5, 3, 3, 6, 9, 9, 7, 5, 4, 7, 11, 12, 10, 7, 6, 9, 14, 16, 14, 11, 8, 11, 17, 20, 19, 15, 12, 14, 21, 26, 25, 21, 17, 18, 26, 32, 33, 28, 23, 24, 32, 41
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 10 2016

Keywords

Comments

Number of partitions of n into parts larger than 1 and congruent to 1 mod 6.

Examples

			a(26) = 2, because we have [19, 7] and [13, 13].

Links

Crossrefs

Cf. A016921, A087897, A109701 (partial sums), A117957, A277210, A277264.

Programs

  • Maple
    N:= 100:
G:= 1/mul(1-x^m,m=7..N,6):
S:= series(G,x,N+1):
seq(coeff(S,x,j),j=0..N); # Robert Israel, Jan 23 2019
  • Mathematica
    CoefficientList[Series[(1 - x)/QPochhammer[x, x^6], {x, 0, 100}], x]

Formula

G.f.: Product_{k>=1} 1/(1 - x^(6*k+1)).
a(n) ~ Pi^(1/6) * Gamma(1/6) * exp(sqrt(n)*Pi/3) / (24*sqrt(3)*n^(13/12)). - Vaclav Kotesovec, Oct 10 2016
Showing 1-3 of 3 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.