A292043 G.f.: Im((i*x; x)_inf), where (a; q)_inf is the q-Pochhammer symbol, i = sqrt(-1).

0, -1, -1, -1, -1, -1, 0, 0, 1, 2, 3, 4, 6, 7, 9, 10, 12, 13, 15, 15, 16, 16, 16, 14, 13, 9, 6, 0, -5, -14, -22, -34, -45, -60, -74, -93, -110, -132, -152, -177, -199, -226, -249, -277, -300, -328, -348, -373, -389, -408, -417, -428, -425, -424, -407, -389, -352

Offset

0,10

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Eric Weisstein's World of Mathematics, q-Pochhammer Symbol.

Formula

(i*x; x)_inf is the g.f. for A292042(n) + i*a(n).

A292042(n)^2 + a(n)^2 = A278420(n). - Vaclav Kotesovec, Sep 08 2017

From Peter Bala, Feb 05 2021: (Start)

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

The 2 X 2 matrix Product_{k >= 1} [1, -x^k; x^k, 1] = [A(x), B(x); -B(x), A(x)], where A(x) is the g.f. of A292042 and B(x) is the g.f. for this sequence.

A(x)^2 + B(x)^2 = Product_{k >= 1} 1 + x^(2*k) = A000009(x^2).

A(x) + B(x) is the g.f. of A278399; B(x) - A(x) is the g.f. of A278400. (End)

Example

Product_{k>=1} (1 - i*x^k) = 1 + (0-1i)*x + (0-1i)*x^2 + (-1-1i)*x^3 + (-1-1i)*x^4 + (-2-1i)*x^5 + (-2+0i)*x^6 + (-3+0i)*x^7 + ...

Maple

N:= 100:
S := convert(series( add( (-1)^(n+1)*x^((n+1)*(2*n+1))/(mul(1 - x^k, k = 1..2*n+1)), n = 0..floor(sqrt(N/2)) ), x, N+1 ), polynom):
seq(coeff(S, x, n), n = 0..N); # Peter Bala, Feb 05 2021

Mathematica

Im[CoefficientList[Series[QPochhammer[I*x, x], {x, 0, 100}], x]] (* Vaclav Kotesovec, Sep 08 2017 *)

Crossrefs

Cf. A278399, A278400, A278420, A292042, A292052.

Sequence in context: A001953 A230748 A078607 * A292052 A347624 A187484

Adjacent sequences: A292040 A292041 A292042 * A292044 A292045 A292046

Keyword

sign

Author

Seiichi Manyama, Sep 08 2017

Status

approved