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A204854 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (1 - x^k) / (1 + x^k). 1
1, 1, -1, 1, -3, 3, -3, 3, -3, 5, -5, 5, -5, 5, -5, 5, -7, 7, -7, 7, -7, 7, -7, 7, -7, 9, -9, 9, -9, 9, -9, 9, -9, 9, -9, 9, -11, 11, -11, 11, -11, 11, -11, 11, -11, 11, -11, 11, -11, 13, -13, 13, -13, 13, -13, 13, -13, 13, -13, 13, -13, 13, -13, 13, -15, 15, -15, 15, -15, 15, -15, 15 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..10000

FORMULA

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

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

EXAMPLE

G.f.: A(x) = 1 + x - x^2 + x^3 - 3*x^4 + 3*x^5 - 3*x^6 + 3*x^7 - 3*x^8 + 5*x^9 +...

where the g.f. is given by the series identities:

(0) A(x) = 1 + x*(1-x)/(1+x) + x^2*(1-x)*(1-x^2)/((1+x)*(1+x^2)) + x^3*(1-x)*(1-x^2)*(1-x^3)/((1+x)*(1+x^2)*(1+x^3)) + x^4*(1-x)*(1-x^2)*(1-x^3)*(1-x^4)/((1+x)*(1+x^2)*(1+x^3)*(1+x^4)) +...

(1) A(x) = 1 + x*(1+x^3)/(1+x) - 3*x^4*(1+x^5)/(1+x) + 5*x^9*(1+x^7)/(1+x) - 7*x^16*(1+x^9)/(1+x) + 9*x^25*(1+x^11)/(1+x) -+...

(2) A(x) = (1 + 2*x - 2*x^4 + 2*x^9 - 2*x^16 + 2*x^25 - 2*x^36 +...)/(1+x).

PROG

(PARI) {a(n)=polcoeff(1+sum(m=1, n, x^m*prod(k=1, m, (1-x^k)/(1+x^k+x*O(x^n)))), n)}

CROSSREFS

Cf. A002448 (Jacobi theta_4), A207641.

Sequence in context: A187469 A112593 A213809 * A113215 A105591 A130497

Adjacent sequences:  A204851 A204852 A204853 * A204855 A204856 A204857

KEYWORD

sign

AUTHOR

Paul D. Hanna, Jan 20 2012

STATUS

approved

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Last modified May 16 09:28 EDT 2021. Contains 343940 sequences. (Running on oeis4.)