OFFSET
0,4
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..5000
Mircea Merca, Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer, Journal of Number Theory, Volume 160, March 2016, Pages 60-75, function s_e(n).
Eric Weisstein's World of Mathematics, q-Polygamma Function, q-Pochhammer Symbol.
FORMULA
a(n)=(1/2)*A015723(n)-(1/2)*sum{k=0..A235963(n)-1, (-1)^A110654(k)*A000005(n-A001318(k))}=A015723(n)-A238131(n).
G.f.: (1/2)*prod(k>=1, 1+x^k ) * sum(k>=1, x^k/(1+x^k) ) - (1/2)*prod(k>=1, 1-x^k) * sum(k>=1, x^k/(1-x^k) ).
G.f.: -(2 * (x; x)_inf * (log(1-x) + psi_x(1)) + (-1; x)_inf * (log(1-x) + psi_x(1-log(-1)/log(x))))/(4*log(x)), where psi_q(z) is the q-digamma function, (a; q)_inf is the q-Pochhammer symbol, log(-1) = i*Pi. - Vladimir Reshetnikov, Nov 21 2016
a(n) ~ 3^(1/4) * log(2) * exp(Pi*sqrt(n/3)) / (4*Pi*n^(1/4)). - Vaclav Kotesovec, May 27 2018
EXAMPLE
a(8)=6 because the partitions of 8 into even number of distinct parts are: 7+1, 6+2 and 5+3.
MAPLE
b:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0,
`if`(n=0, [1, 0$3], b(n, i-1)+`if`(i>n, 0, (p->
[p[2], p[1], p[4]+p[2], p[3]+p[1]])(b(n-i, i-1)))))
end:
a:= n-> b(n$2)[3]:
seq(a(n), n=0..60); # Alois P. Heinz, Dec 27 2015
MATHEMATICA
max = 50; s = (1/2)*Product[1+x^k, {k, 1, max}]*Sum[x^k/(1+x^k), {k, 1, max}] - (1/2)*Product[1-x^k, {k, 1, max}]*Sum[x^k/(1-x^k), {k, 1, max}] + O[x]^(max+1); CoefficientList[s, x] (* Jean-François Alcover, Dec 27 2015 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Mircea Merca, Feb 18 2014
STATUS
approved