cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A349846 Expansion of -(1 - 8*x) / sqrt(1 - 4*x).

Original entry on oeis.org

-1, 6, 10, 28, 90, 308, 1092, 3960, 14586, 54340, 204204, 772616, 2939300, 11232648, 43088200, 165815280, 639859770, 2475036900, 9593714460, 37255818600, 144915581580, 564514356120, 2201964031800, 8599360982160, 33619842137700, 131570223027048, 515366318553912
Offset: 0

Views

Author

Jianing Song, Dec 01 2021

Keywords

Comments

Sum_{n>=0} (-a(n)/(-4)^n) is the Cauchy product of Sum_{n>=0} (-A349844(n)/(-8)^n) with itself.

Examples

			a(1) = binomial(0,0) * (4 + 2/1) = 6;
a(2) = binomial(2,1) * (4 + 2/2) = 10;
a(3) = binomial(4,2) * (4 + 2/3) = 28;
a(4) = binomial(6,3) * (4 + 2/4) = 90.

Links

Crossrefs

Cf. A000984, A349844, A349835, A349847.

Programs

  • PARI
    a(n) = if(n, binomial(2*(n-1),n-1) * (4 + 2/n), -1)

Formula

For n > 0, a(n) = 8*binomial(2*(n-1),n-1) - binomial(2*n,n) = binomial(2*(n-1),n-1) * (4 + 2/n).
a(n) ~ 4^n * (1/sqrt(Pi*n)).

A349847 Expansion of (1 + 8*x) / sqrt(1 - 4*x).

Original entry on oeis.org

1, 10, 22, 68, 230, 812, 2940, 10824, 40326, 151580, 573716, 2183480, 8347612, 32033848, 123321400, 476050320, 1842020550, 7142249340, 27743985060, 107946346200, 420608639220, 1641030105000, 6410161959240, 25066222437360, 98115049503900, 384391435902552
Offset: 0

Views

Author

Jianing Song, Dec 01 2021

Keywords

Comments

Sum_{n>=0} (a(n)/(-4)^n) is the Cauchy product of Sum_{n>=0} (-A349845(n)/8^n) with itself.

Examples

			a(1) = binomial(0,0) * (12 - 2/1) = 10;
a(2) = binomial(2,1) * (12 - 2/2) = 22;
a(3) = binomial(4,2) * (12 - 2/3) = 68;
a(4) = binomial(6,3) * (12 - 2/4) = 230.

Links

Crossrefs

Cf. A000984, A349845, A349835, A349846.

Programs

  • Mathematica
    CoefficientList[Series[(1+8x)/Sqrt[1-4x],{x,0,30}],x] (* Harvey P. Dale, Jun 08 2023 *)
  • PARI
    a(n) = if(n, binomial(2*(n-1),n-1) * (12 - 2/n), 1)

Formula

For n > 0, a(n) = 8*binomial(2*(n-1),n-1) + binomial(2*n,n) = binomial(2*(n-1),n-1) * (12 - 2/n).
a(n) ~ 4^n * (3/sqrt(Pi*n)).

A349834 Expansion of sqrt(1 + 4*x)/(1 - 4*x).

Original entry on oeis.org

1, 6, 22, 92, 358, 1460, 5756, 23288, 92294, 372036, 1478420, 5947272, 23671516, 95102088, 378922552, 1521039088, 6064766662, 24329781988, 97059838372, 389194630888, 1553243997172, 6226104229528, 24855484384072, 99604902663568, 397733491426972
Offset: 0

Views

Author

Jianing Song, Dec 01 2021

Keywords

Comments

Let b(n) = a(n)/4^n, {b(n)} = {1, 3/2, 11/8, 23/16, 179/128, 365/256, 1439/1024, ...}. Since a(n) >= 4^n, Sum_{n>=0} b(n) is divergent. Let c(n) = A349835(n)/(-4)^n, {c(n)} = {1, -3/2, 7/8, -11/16, 75/128, -133/256, 483/1024, ...}. Since |c(n)| ~ 2/sqrt(Pi*n) and |c(n+1)|/|c(n)| = ((4*n+3)*(2*n-1)) / ((4*n-1)*(2*n+2)) < 1, Sum_{n>=0} c(n) is conditionally convergent by Leibniz's criterion. Note that Sum_{n>=0} b(n)*x^n = sqrt(1 + x)/(1 - x), Sum_{n>=0} c(n)*x^n = (1 - x)/sqrt(1 + x), hence the Cauchy product of Sum_{n>=0} b(n) and Sum_{n>=0} c(n) is 1 + 0 + 0 + .... {b(n)} and {c(n)} serve as an example such that the Cauchy product of a divergent series and a conditionally convergent series can be absolutely convergent.

Examples

			Let C(n) denote the Catalan numbers.
a(0) = 2^0 = 1;
a(1) = 2^2 + 2^1 * C(0) = 6;
a(2) = 2^4 + 2^3 * C(0) - 2^1 * C(1) = 22;
a(3) = 2^6 + 2^5 * C(0) - 2^3 * C(1) + 2^1 * C(2) = 92;
a(4) = 2^8 + 2^7 * C(0) - 2^5 * C(1) + 2^3 * C(2) - 2^1 * C(3) = 358.

Links

Crossrefs

Cf. A000108, A349835.

Programs

  • PARI
    C(n) = binomial(2*n, n)/(n+1)
a(n) = 2^(2*n) + sum(k=0, n-1, (-1)^k * 2^(2*n-1-2*k) * C(k))

Formula

a(n) = 2^(2*n) + (Sum_{k=0..n-1} (-1)^k * 2^(2*n-1-2*k) * CatalanNumber(k)).
a(n) = 2^(2*n + 1/2) - ((-1)^n * CatalanNumber(n) * hypergeom([1, n + 1/2], [n + 2], -1)) / 2. - Peter Luschny, Dec 03 2021
D-finite with recurrence n*a(n) -6*a(n-1) +8*(-2*n+3)*a(n-2)=0. - R. J. Mathar, Jul 27 2022
Showing 1-3 of 3 results.

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