login
G.f.: Re(1/(i*x; x)_inf), where (a; q)_inf is the q-Pochhammer symbol, i = sqrt(-1).
5

%I #29 Jan 19 2021 21:53:38

%S 1,0,-1,-1,-1,-1,-2,-1,0,0,0,1,2,3,3,4,6,6,5,6,7,6,5,5,5,3,0,-2,-3,-6,

%T -11,-13,-14,-19,-24,-27,-29,-33,-38,-40,-40,-43,-47,-46,-43,-43,-43,

%U -38,-30,-26,-22,-12,1,11,20,36,56,71,85,106,130,149,166,190,217

%N G.f.: Re(1/(i*x; x)_inf), where (a; q)_inf is the q-Pochhammer symbol, i = sqrt(-1).

%H Seiichi Manyama, <a href="/A292136/b292136.txt">Table of n, a(n) for n = 0..1000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/q-PochhammerSymbol.html">q-Pochhammer Symbol</a>.

%F 1/( i*x; x)_inf is the g.f. for a(n) + i*A292137(n).

%F 1/(-i*x; x)_inf is the g.f. for a(n) + i*A292138(n).

%F From _Peter Bala_, Jan 19 2021: (Start)

%F 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.

%F G.f.: Sum_{n >= 0} (-1)^n * x^(2*n)/Product_{k = 1..2*n} (1 - x^k). (End)

%e 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 + ...

%e 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 + ...

%e From _Peter Bala_, Jan 19 2021: (Start)

%e The number of partitions of n = 13 into an even number of parts is:

%e # parts (2*k) 2 4 6 8 10 12

%e # partitions 6 18 14 7 3 1

%e Hence a(13) = Sum (-1)^k = -6 + 18 - 14 + 7 - 3 + 1 = 3. (End)

%p N:= 100:

%p S := convert(series( add( (-1)^n*x^(2*n)/(mul(1 - x^k,k = 1..2*n)), n = 0..N ), x, N+1 ), polynom):

%p seq(coeff(S, x, n), n = 0..N); # _Peter Bala_, Jan 15 2021

%t Re[CoefficientList[Series[1/QPochhammer[I*x, x], {x, 0, 100}], x]] (* _Vaclav Kotesovec_, Sep 17 2017 *)

%Y Cf. A292042, A292043, A292137, A292138.

%K sign,look

%O 0,7

%A _Seiichi Manyama_, Sep 09 2017