OFFSET
0,7
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, q-Pochhammer Symbol.
FORMULA
1/( i*x; x)_inf is the g.f. for a(n) + i*A292137(n).
1/(-i*x; x)_inf is the g.f. for a(n) + i*A292138(n).
From Peter Bala, Jan 19 2021: (Start)
a(n) = Sum (-1)^k, where the sum is over all integer partitions of n into an even number of parts and 2*k is the number of parts in a partition. An example is given below.
G.f.: Sum_{n >= 0} (-1)^n * x^(2*n)/Product_{k = 1..2*n} (1 - x^k). (End)
EXAMPLE
Product_{k>=1} 1/(1 - i*x^k) = 1 + (0+1i)*x + (-1+1i)*x^2 + (-1+0i)*x^3 + (-1+0i)*x^4 + (-1+0i)*x^5 + (-2-1i)*x^6 + (-1-2i)*x^7 + ...
Product_{k>=1} 1/(1 + i*x^k) = 1 + (0-1i)*x + (-1-1i)*x^2 + (-1+0i)*x^3 + (-1+0i)*x^4 + (-1+0i)*x^5 + (-2+1i)*x^6 + (-1+2i)*x^7 + ...
From Peter Bala, Jan 19 2021: (Start)
The number of partitions of n = 13 into an even number of parts is:
# parts (2*k) 2 4 6 8 10 12
# partitions 6 18 14 7 3 1
Hence a(13) = Sum (-1)^k = -6 + 18 - 14 + 7 - 3 + 1 = 3. (End)
MAPLE
N:= 100:
S := convert(series( add( (-1)^n*x^(2*n)/(mul(1 - x^k, k = 1..2*n)), n = 0..N ), x, N+1 ), polynom):
seq(coeff(S, x, n), n = 0..N); # Peter Bala, Jan 15 2021
MATHEMATICA
Re[CoefficientList[Series[1/QPochhammer[I*x, x], {x, 0, 100}], x]] (* Vaclav Kotesovec, Sep 17 2017 *)
CROSSREFS
KEYWORD
sign,look
AUTHOR
Seiichi Manyama, Sep 09 2017
STATUS
approved