A261772 Expansion of Product_{k>=1} (1 + x^k) / (1 + x^(10*k)).

1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 9, 11, 14, 16, 20, 24, 28, 33, 40, 46, 54, 64, 74, 86, 100, 115, 133, 154, 176, 202, 231, 263, 300, 342, 388, 440, 499, 563, 636, 718, 808, 909, 1022, 1146, 1284, 1439, 1608, 1797, 2006, 2236, 2490, 2772, 3081, 3422, 3800, 4212

Offset: 0

Author

Vaclav Kotesovec, Aug 31 2015

Keywords

• nonn

Comments

a(n) is the number of partitions of n into distinct parts where no part is a multiple of 10.

In general, if m > 1 and g.f. = Product_{k>=1} (1 + x^k)/(1 + x^(m*k)), then a(n) ~ exp(Pi*sqrt((m-1)*n/(3*m))) * (m-1)^(1/4) / (2^(3/2) * 3^(1/4) * m^(1/4) * n^(3/4)) * (1 - (3*sqrt(3*m)/(8*Pi*sqrt(m-1)) + (m-1)^(3/2)*Pi/(48*sqrt(3*m))) / sqrt(n)). - Vaclav Kotesovec, Aug 31 2015, extended Jan 21 2017

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A145707.

Cf. A000700 (m=2), A003105 (m=3), A070048 (m=4), A096938 (m=5), A261770 (m=6), A097793 (m=7), A261771 (m=8), A112193 (m=9).

Column k=10 of A290307.

Programs

Maple

b:= proc(n, i) option remember;  local r;
      `if`(2*n>i*(i+1)-(j-> 10*j*(j+1))(iquo(i, 10, 'r')), 0,
      `if`(n=0, 1, b(n, i-1)+`if`(i>n or r=0, 0, b(n-i, i-1))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..80);  # Alois P. Heinz, Aug 31 2015

Mathematica

nmax = 50; CoefficientList[Series[Product[(1 + x^k) / (1 + x^(10*k)), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ exp(Pi*sqrt(3*n/10)) * 3^(1/4) / (2^(7/4) * 5^(1/4) * n^(3/4)) * (1 - (sqrt(15)/(4*Pi*sqrt(2)) + 3*Pi*sqrt(3)/(16*sqrt(10))) / sqrt(n)). - Vaclav Kotesovec, Aug 31 2015, extended Jan 21 2017

G.f.: Product_{k>=1} (1 - x^(20*k-10))/(1 - x^(2*k-1)). - Ilya Gutkovskiy, Dec 07 2017