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

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, -11, -13, -14, -19, -24, -27, -29, -33, -38, -40, -40, -43, -47, -46, -43, -43, -43, -38, -30, -26, -22, -12, 1, 11, 20, 36, 56, 71, 85, 106, 130, 149, 166, 190, 217

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

Cf. A292042, A292043, A292137, A292138.

Sequence in context: A344858 A110174 A022909 * A032239 A057094 A284938

Adjacent sequences: A292133 A292134 A292135 * A292137 A292138 A292139

Keyword

sign,look

Author

Seiichi Manyama, Sep 09 2017

Status

approved