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.

A009564 E.g.f. sin(x^2)/2, coefficients of x^(4*n + 2).

Original entry on oeis.org

1, -60, 15120, -8648640, 8821612800, -14079294028800, 32382376266240000, -101421602465863680000, 415017197290314178560000, -2149789081963827444940800000, 13750050968240640337841356800000, -106425394494182556214892101632000000, 980390734080409707851586040233984000000
Offset: 0

Views

Author

R. H. Hardin

Keywords

Links

Crossrefs

Cf. A001813, A103639, A024343, A075069.

Programs

  • Magma
    [(-1)^n*Factorial(2+4*n)/(2*Factorial(1+2*n)): n in [0..20]]; // Vincenzo Librandi, Dec 22 2015
  • Maple
    seq(i!*coeff(series(sin(x^2)/2,x,4*i+4),x,i),i=2..54,4); # Peter Luschny, Dec 14 2012
  • Mathematica
    nmax = 12; coes = CoefficientList[ Series[ Sin[x^2]/2, {x, 0, 4*nmax + 2}], x]; a[n_] := coes[[4*n + 3]]*(4*n + 2)!; Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Dec 14 2012 *)
Table[(-1)^n (2 + 4 n)!/(2 (1 + 2 n)!), {n, 0, 25}] (* Vincenzo Librandi, Dec 22 2015 *)
  • PARI
    a(n) = (-1)^n*(2+4*n)!/(2*(1+2*n)!); \\ Altug Alkan, Dec 22 2015
  • Sage
    def A009564(n):
    return falling_factorial(4*n+2,2*n+1)/(2*(-1)^n)
[A009564(n) for n in (0..12)]  # Peter Luschny, Dec 14 2012

Formula

a(n) = (-1)^n*(2+4*n)!/(2*(1+2*n)!) = (-1)^n*A001813(2*n+1)/2. - Robert Israel, Dec 21 2015
From Amiram Eldar, Sep 02 2025: (Start)
a(n) = A024343(n)/2.
Sum_{n>=0} 1/a(n) = sqrt(2*Pi) * (cos(1/4) * FresnelC(1/sqrt(2*Pi)) + sin(1/4) * FresnelS(1/sqrt(2*Pi))), where FresnelC(x) and FresnelS(x) are the Fresnel integrals C(x) and S(x), respectively.
Sum_{n>=0} (-1)^n/a(n) = (sqrt(Pi)/2) * (exp(1/4) * erf(1/2) + erfi(1/2) / exp(1/4)). (End)

Extensions

Extended with signs Mar 1997
Definition corrected and terms a(10)-a(12) from Peter Luschny, Dec 14 2012

A262206 Product of prime(n) consecutive numbers starting from n.

Original entry on oeis.org

2, 24, 2520, 604800, 54486432000, 53353114214400, 35905578804006912000, 80018147048929689600000, 203939450748460387344384000000, 1441310123089178548721360295690240000000, 9218619547278385997621820451234775040000000
Offset: 1

Views

Author

Altug Alkan, Sep 15 2015

Keywords

Comments

a(n) is always divisible by A039716(n).

Examples

			For n=1, a(1) = 1*2 = 2.
For n=2, a(2) = 2*3*4 = 24.
For n=3, a(3) = 3*4*5*6*7 = 2520.
For n=4, a(4) = 4*5*6*7*8*9*10 = 604800.

Crossrefs

Cf. A075069: product of prime(n) consecutive numbers starting from prime(n).
Cf. A014688, A039716, A075068, A262204.

Programs

Formula

a(n) = (prime(n) + n - 1)! / (n-1)!.

A262204 a(n) = (2*prime(n))! / prime(n)!.

Original entry on oeis.org

12, 120, 30240, 17297280, 28158588057600, 64764752532480000, 830034394580628357120000, 4299578163927654889881600000, 212850788988365112429784203264000000, 265847614191284935213187014536606662000640000000
Offset: 1

Views

Author

Altug Alkan, Sep 15 2015

Keywords

Comments

Inspired by simplicity of sequence formula that is (p + p)!/p! where p is n-th prime number.

Examples

			For n=1, a(n) = (2*prime(n))! / prime(n)! = 4!/2! = 3*4 = 12.
For n=2, a(n) = (2*prime(n))! / prime(n)! = 6!/3! = 4*5*6 = 120.
For n=3, a(n) = (2*prime(n))! / prime(n)! = 10!/5! = 6*7*8*9*10 = 30240.

Crossrefs

Cf. A001813, A039716, A100484, A075069, A262206.

Programs

  • Magma
    [Factorial(NthPrime(n)+NthPrime(n)) / Factorial(NthPrime(n)): n in [1..10]]; // Vincenzo Librandi, Sep 16 2015
  • PARI
    a(n) = (2*prime(n))!/prime(n)!;
vector(10, n, a(n))

Formula

a(n) = 2 * A075069(n).
a(n) = A001813(prime(n)). - Michel Marcus, Sep 20 2015
a(n) mod A039716(n) = 0.
Showing 1-3 of 3 results.

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