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A292136 G.f.: Re(1/(i*x; x)_inf), where (a; q)_inf is the q-Pochhammer symbol, i = sqrt(-1).

%I A292136 #30 Feb 16 2025 08:33:51
%S A292136 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 A292136 -11,-13,-14,-19,-24,-27,-29,-33,-38,-40,-40,-43,-47,-46,-43,-43,-43,
%U A292136 -38,-30,-26,-22,-12,1,11,20,36,56,71,85,106,130,149,166,190,217
%N A292136 G.f.: Re(1/(i*x; x)_inf), where (a; q)_inf is the q-Pochhammer symbol, i = sqrt(-1).
%H A292136 Seiichi Manyama, <a href="/A292136/b292136.txt">Table of n, a(n) for n = 0..1000</a>
%H A292136 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/q-PochhammerSymbol.html">q-Pochhammer Symbol</a>.
%F A292136 1/( i*x; x)_inf is the g.f. for a(n) + i*A292137(n).
%F A292136 1/(-i*x; x)_inf is the g.f. for a(n) + i*A292138(n).
%F A292136 From _Peter Bala_, Jan 19 2021: (Start)
%F A292136 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 A292136 G.f.: Sum_{n >= 0} (-1)^n * x^(2*n)/Product_{k = 1..2*n} (1 - x^k). (End)
%e A292136 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 A292136 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 A292136 From _Peter Bala_, Jan 19 2021: (Start)
%e A292136 The number of partitions of n = 13 into an even number of parts is:
%e A292136 # parts (2*k)   2    4    6   8   10   12
%e A292136 # partitions    6   18   14   7    3    1
%e A292136 Hence a(13) = Sum (-1)^k = -6 + 18 - 14 + 7 - 3 + 1 = 3. (End)
%p A292136 N:= 100:
%p A292136 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 A292136 seq(coeff(S, x, n), n = 0..N); # _Peter Bala_, Jan 15 2021
%t A292136 Re[CoefficientList[Series[1/QPochhammer[I*x, x], {x, 0, 100}], x]] (* _Vaclav Kotesovec_, Sep 17 2017 *)
%Y A292136 Cf. A292042, A292043, A292137, A292138.
%K A292136 sign,look
%O A292136 0,7
%A A292136 _Seiichi Manyama_, Sep 09 2017