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 1-5 of 5 results.

A384166 a(n) = Product_{k=0..n-1} (3*n+4*k).

Original entry on oeis.org

1, 3, 60, 1989, 92160, 5486535, 399072960, 34298042625, 3400783626240, 382128386114475, 47986411423104000, 6659996213472126525, 1012334387351519232000, 167253493686752981883375, 29842935065036371998720000, 5719198821953333723419037625, 1171620424982972483984424960000
Offset: 0

Views

Author

Seiichi Manyama, May 21 2025

Keywords

Links

Crossrefs

Cf. A384164, A384165.
Cf. A303487.

Programs

  • Magma
    [1] cat  [&*[(3*n + 4*k): k in [0..n-1]]: n in [1..16]]; // Vincenzo Librandi, May 22 2025
  • Mathematica
    a[n_]:=Product[(3*n+4*k),{k,0,n-1}];Table[a[n],{n,0,15}] (* Vincenzo Librandi, May 22 2025 *)
  • PARI
    a(n) = prod(k=0, n-1, 3*n+4*k);
  • Python
    from math import prod
def A384166(n): return prod(3*n+i for i in range(0,n<<2,4)) # Chai Wah Wu, May 21 2025
  • Sage
    def a(n): return 4^n*rising_factorial(3*n/4, n)

Formula

a(n) = 4^n * RisingFactorial(3*n/4,n).
a(n) = n! * [x^n] 1/(1 - 4*x)^(3*n/4).
a(n) = (3/7) * 4^n * n! * binomial(7*n/4,n) for n > 0.

A385163 Let p = A002145(n) be the n-th prime == 3 (mod 4); a(n) is the multiplicative order of 1+-i modulo p in Gaussian integers.

Original entry on oeis.org

8, 24, 40, 72, 88, 40, 56, 184, 232, 264, 280, 312, 328, 408, 424, 56, 520, 552, 120, 648, 664, 712, 760, 792, 840, 296, 904, 952, 200, 1048, 1080, 376, 408, 1240, 120, 1384, 1432, 1464, 1512, 1528, 1672, 344, 584, 1768, 1848, 1864, 1912, 1944, 1960, 664, 2008, 2088, 2184, 2248, 456
Offset: 1

Views

Author

Jianing Song, Jun 20 2025

Keywords

Comments

Also, a(n) is the multiplicative order of the matrix [1,-1;1,1] or [1,1;-1,1] modulo p.
Note that (1+-i)^4 = -4. Since (1+-i)^n is a real number if and only if n is divisible by 4, we have a(n) = 4*ord(-4,p), where ord(a,p) is the multiplicative order of a modulo p.

Examples

			For A002145(4) = 19: Since (1+i)^(4k) = (-4)^k, we have (1+i)^72 == 1 (mod 19), and 72 is the smallest such exponent. Hence a(4) = 72.

Links

Crossrefs

Cf. A002145, A384164 ({a(n)/8}), A385165 (multiplicative order of 2+-i).

Programs

  • PARI
    forprime(p=3, 1e3, if(p%4==3, print1(4*znorder(Mod(-4,p)), ", ")))

A384165 a(n) = Product_{k=0..n-1} (3*n+2*k).

Original entry on oeis.org

1, 3, 48, 1287, 48384, 2340135, 138378240, 9672183675, 780151357440, 71322093677835, 7287813911347200, 823100991923184975, 101819334240239616000, 13690816766440373134575, 1988199345147516813312000, 310120801435080997013527875, 51709528644340997758648320000
Offset: 0

Views

Author

Seiichi Manyama, May 21 2025

Keywords

Links

Crossrefs

Cf. A384164, A384166.

Programs

  • Magma
    [1] cat  [&*[(3*n + 2*k): k in [0..n-1]]: n in [1..16]]; // Vincenzo Librandi, May 22 2025
  • Mathematica
    a[n_]:=Product[(3*n+2*k),{k,0,n-1}]; Table[a[n],{n,0,15}] (* Vincenzo Librandi, May 22 2025 *)
  • PARI
    a(n) = prod(k=0, n-1, 3*n+2*k);
  • Python
    from math import prod
def A384165(n): return prod(3*n+i for i in range(0,n<<1,2)) # Chai Wah Wu, May 21 2025
  • Sage
    def a(n): return 2^n*rising_factorial(3*n/2, n)

Formula

a(n) = 2^n * RisingFactorial(3*n/2,n).
a(n) = n! * [x^n] 1/(1 - 2*x)^(3*n/2).
a(n) = (3/5) * 2^n * n! * binomial(5*n/2,n) for n > 0.

A384262 a(n) = Product_{k=0..n-1} (3*n+k-2).

Original entry on oeis.org

1, 1, 20, 504, 17160, 742560, 39070080, 2422728000, 173059286400, 13995229248000, 1264020397516800, 126115611484262400, 13776096431889792000, 1635195634511530291200, 209574860127295703040000, 28844656968251942737920000, 4243193364951971128258560000, 664387519844376163893657600000
Offset: 0

Views

Author

Seiichi Manyama, May 23 2025

Keywords

Crossrefs

Cf. A061924, A384164, A384263.
Cf. A335647.

Programs

  • PARI
    a(n) = prod(k=0, n-1, 3*n+k-2);
  • Python
    from sympy import rf
def a(n): return rf(3*n-2, n)
  • Sage
    def a(n): return rising_factorial(3*n-2, n)

Formula

a(n) = RisingFactorial(3*n-2,n).
a(n) = n! * [x^n] 1/(1 - x)^(3*n-2).
a(n) = n! * binomial(4*n-3,n).
D-finite with recurrence 3*(3*n-4)*(3*n-5)*a(n) -8*(4*n-5)*(4*n-3)*(2*n-3)*a(n-1)=0. - R. J. Mathar, May 26 2025

A384263 a(n) = Product_{k=0..n-1} (3*n+k-1).

Original entry on oeis.org

1, 2, 30, 720, 24024, 1028160, 53721360, 3315312000, 235989936000, 19033511777280, 1715456253772800, 170866312333516800, 18638248113733248000, 2209723830420986880000, 282926061171849199104000, 38906746608339829739520000, 5719086709283091520696320000, 894889312443445445244518400000
Offset: 0

Views

Author

Seiichi Manyama, May 23 2025

Keywords

Crossrefs

Cf. A061924, A384164, A384262.

Programs

  • Mathematica
    a[n_]:=n!*Binomial[4*n-2,n]; Array[a,18,0] (* Stefano Spezia, Sep 04 2025 *)
  • PARI
    a(n) = prod(k=0, n-1, 3*n+k-1);
  • Python
    from sympy import rf
def a(n): return rf(3*n-1, n)
  • Sage
    def a(n): return rising_factorial(3*n-1, n)

Formula

a(n) = RisingFactorial(3*n-1,n).
a(n) = n! * [x^n] 1/(1 - x)^(3*n-1).
a(n) = n! * binomial(4*n-2,n).
From Stefano Spezia, Sep 04 2025: (Start)
E.g.f.: (1 + 3*hypergeom([-1/4, 1/4, 1/2], [-1/3, 1/3], 2^8*x/3^3])/4.
a(n) ~ 2^(8*n-7)*3^(-3*n-1/2)*exp(-n)*n^(n-1)*(144*n - 13). (End)
Showing 1-5 of 5 results.

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