A318570 -id:A318570 - cp's OEIS frontend

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

Seiichi Manyama, Apr 12 2019

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

			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)

Convolution inverse of A318570.

Cf. A002448, A307521, A307522.

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

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 *)

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

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, n-k^2). - Peter Bala, Dec 31 2024

A307521 Expansion of Product_{k>=1} ((1 + x)^k + x^k)/((1 + x)^k - x^k).

Original entry on oeis.org

1, 2, 2, 2, 0, 2, -2, 2, 0, -4, 8, -4, -26, 112, -288, 560, -832, 782, 274, -3378, 9424, -17498, 21182, -2154, -78180, 284594, -700018, 1381802, -2250316, 2877674, -2172870, -1955998, 12715122, -33812990, 67322842, -108956110, 139447006, -110023870, -83188990, 651268018

Offset: 0

Seiichi Manyama, Apr 12 2019

sign

Cf. A002448, A307520, A307522, A318570.

m = 39; 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 *)

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

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

A307522 Expansion of Product_{k>=1} ((1 + x)^k - x^k)/((1 + x)^k + x^k).

Original entry on oeis.org

1, -2, 2, -2, 4, -10, 22, -42, 72, -116, 188, -332, 662, -1432, 3148, -6736, 13784, -26894, 50254, -90782, 160856, -285230, 518170, -983710, 1964800, -4090002, 8705322, -18582722, 39219572, -81148034, 163946630, -323136562, 622125982, -1173528562, 2179230066

Offset: 0

Seiichi Manyama, Apr 12 2019

Cf. A002448, A103198, A307520, A307521, A318570.

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

m = 34; 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 *)

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

G.f.: theta_4(x/(1 + x)), where theta_4() is the Jacobi theta function.
From Peter Bala, Dec 31 2024: (Start)
For n >= 1, a(n) = 2 * (-1)^n * Sum_{k = 1..floor(sqrt(n))} binomial(n-1, n-k^2).
For n >= 1, |a(n)| = 2 * A103198(n). (End)

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

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A307521 Expansion of Product_{k>=1} ((1 + x)^k + x^k)/((1 + x)^k - x^k).

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A307522 Expansion of Product_{k>=1} ((1 + x)^k - x^k)/((1 + x)^k + x^k).

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