A307520 Expansion of Product_{k>=1} ((1 - x)^k - x^k)/((1 - x)^k + x^k).

1, -2, -2, -2, 0, 6, 18, 38, 68, 108, 148, 148, -2, -552, -2004, -5280, -11960, -24590, -46990, -84254, -141476, -218654, -295794, -295374, 3540, 1148110, 4378918, 12332318, 30327568, 68633710, 146303026, 297349758, 580052778, 1089382158, 1969425038, 3413378318, 5614375970

Offset

0,2

Comments

Let A(x) = 1 - 2*x - 2*x^2 - 2*x^3 + 6*x^5 + 18*x^6 + ... denote the g.f. of the sequence. Let u denote the lower triangular matrix with 0's on the main diagonal and 1's in every position below the main diagonal. Then the array Sum_{n in Z} (-1)^n * u^(n^2) is equal to the Riordan array (A(x), x). - Peter Bala, Dec 31 2024

Links

Table of n, a(n) for n=0..36.

Formula

G.f.: theta_4(x/(1 - x)), where theta_4() is the Jacobi theta function.

For n >= 1, a(n) = 2 * Sum_{k=1..floor(sqrt(n))} (-1)^k * binomial(n-1, k^2-1). - Peter Bala, Dec 31 2024

From Ridouane Oudra, Nov 23 2025: (Start) For n >=1 we have :

a(n) = 2*Sum_{k=1..n} (-1)^k*binomial(n,k)*floor(sqrt(k)).

a(n) = 2*Sum_{k=1..n} (-1)^k*binomial(n+1,k+1)*A022554(k).

a(n) = (2/n)*Sum_{k=1..floor(sqrt(n))} (-1)^k*k^2*binomial(n,k^2).

a(n) = a(n-1) + 2*Sum_{k=1..floor(sqrt(n))} (-1)^k*binomial(n-2,k^2-2).

a(n) = - Sum_{k=0..n-1} A318570(n-k)*a(k), with a(0) = 1. (End)

Example

From Peter Bala, Dec 31 2024: (Start)
With the array u as defined above, the lower triangular array Sum_{n = -2..2} (-1)^n * u^(n^2) = I - 2*u + 2*u^4 begins
   1;
  -2,   1;
  -2,  -2,  1;
  -2,  -2, -2,  1;
   0,  -2, -2, -2,  1;
   6,   0, -2, -2, -2,  1;
  18,   6,  0, -2, -2, -2,  1;
  38,  18,  6,  0, -2, -2, -2,  1;
  68,  38, 18,  6,  0, -2, -2, -2,  1; (End)

Maple

a:= n-> 2*add((-1)^k*binomial(n-1, k^2-1), k=1..floor(sqrt(n))):
print(1, seq(a(n), n=1..40)); # Peter Bala, Dec 31 2024

Mathematica

m = 36; CoefficientList[Series[Product[((1 - x)^k - x^k)/((1 - x)^k + x^k), {k, 1, m}], {x, 0, m}], x] (* Amiram Eldar, May 14 2021 *)

Prog

(PARI) my(N=66, x='x+O('x^N)); Vec(prod(k=1, N, ((1-x)^k-x^k)/((1-x)^k+x^k)))

Crossrefs

Convolution inverse of A318570.

Cf. A002448, A307521, A307522, A022554.

Sequence in context: A136663 A165490 A298819 * A265648 A181230 A262372

Adjacent sequences: A307517 A307518 A307519 * A307521 A307522 A307523

Keyword

sign,easy

Author

Seiichi Manyama, Apr 12 2019

Status

approved