OFFSET
0,4
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
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
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]
CROSSREFS
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, Aug 31 2015
STATUS
approved