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 1-6 of 6 results.

A324569 Decimal expansion of a constant related to the asymptotics of A306729.

Original entry on oeis.org

6, 2, 1, 4, 3, 9, 8, 6, 9, 2, 3, 3, 4, 5, 2, 9, 0, 2, 5, 5, 4, 8, 9, 7, 4, 5, 4, 1, 7, 3, 5, 7, 6, 6, 7, 7, 8, 4, 6, 3, 1, 9, 8, 2, 2, 0, 7, 4, 9, 3, 4, 9, 5, 1, 8, 5, 8, 9, 0, 5, 0, 2, 6, 1, 5, 9, 6, 1, 8, 1, 6, 2, 6, 5, 4, 4, 8, 2, 2, 1, 7, 1, 8, 5, 0, 4, 0, 0, 2, 0, 8, 7, 4, 6, 4, 9, 0, 1, 9, 9, 0, 9, 9, 1, 9, 1
Offset: 2

Views

Author

Vaclav Kotesovec, Mar 07 2019

Keywords

Examples

			62.1439869233452902554897454173576677846319822074934951858905026159618...

Crossrefs

Cf. A306729.

Formula

Equals limit_{n->infinity} A306729(n) / (2^(n^2/2 + 2*n) * Pi^(n^2/2 + n) * n^(2*n^3/3 + 2*n^2 + 11*n/6 + 5/2) / exp(8*n^3/9 + 2*n^2 + n)).

A217757 Product_{i=0..n} (i! + 1).

Original entry on oeis.org

2, 4, 12, 84, 2100, 254100, 183206100, 923541950100, 37238134969982100, 13513011656042074430100, 49036030210457135734021310100, 1957361459740805606124917565020990100, 937579272951542930363610919638075856505150100
Offset: 0

Views

Author

Jon Perry, Mar 23 2013

Keywords

Links

Crossrefs

Cf. A000142, A000178, A054640, A082480, A258325, A260231, A306729.

Programs

  • JavaScript
    function factorial(n) {
var i,c=1;
for (i=2;i<=n;i++) c*=i;
return c;
}
a=2;
for (j=1;j<10;j++) {
a*=(factorial(j)+1);
document.write(a+", ");
}
  • Maple
    a:= proc(n) a(n):= `if`(n=0, 2, a(n-1)*(n!+1)) end:
seq(a(n), n=0..14);  # Alois P. Heinz, May 20 2013
  • Mathematica
    Table[Product[i!+1,{i,0,n}],{n,0,12}]  (* Geoffrey Critzer, May 04 2013 *)
Rest[FoldList[Times,1,Range[0,15]!+1]] (* Harvey P. Dale, May 28 2013 *)

Formula

a(n) ~ c * A000178(n), where c = A238695 = Product_{k>=0} (1 + 1/k!) = 7.364308272367257256372772509631... . - Vaclav Kotesovec, Jul 20 2015

A203482 a(n) = Product_{1 <= i < j <= n} (i! + j!).

Original entry on oeis.org

1, 3, 168, 3276000, 877449500928000, 207244701437748852512194560000, 4000516840149319128119305958853265913416777728000000, 796608816253064941944831363792070377592412324940256242675178274726476775424000000000
Offset: 1

Views

Author

Clark Kimberling, Jan 03 2012

Keywords

Comments

Each term divides its successor, as in A203483.
See A093883 for a guide to related sequences.

Links

Crossrefs

Cf. A000142, A203483, A203510, A306729, A325052.

Programs

  • Magma
    [(&*[(&*[Factorial(j) + Factorial(k): k in [1..j]])/(2*Factorial(j)): j in [1..n]]): n in [1..20]]; // G. C. Greubel, Aug 29 2023
  • Maple
    a:= n-> mul(mul(i!+j!, i=1..j-1), j=2..n):
seq(a(n), n=1..10);  # Alois P. Heinz, Jul 23 2017
  • Mathematica
    (* First program *)
f[j_]:= j!; z = 10;
v[n_]:= Product[Product[f[k] + f[j], {j, k-1}], {k, 2, n}]
d[n_]:= Product[(i-1)!, {i, n}]  (* A000178 *)
Table[v[n], {n, z}]              (* A203482 *)
Table[v[n+1]/v[n], {n, z-1}]     (* A203483 *)
Table[v[n]/d[n], {n, 10}]        (* A203510 *)
(* Second program *)
Table[Product[j!+k!, {j,n}, {k,j-1}], {n,15}] (* G. C. Greubel, Aug 29 2023 *)
  • SageMath
    [product(product(factorial(j) + factorial(k) for k in range(1,j)) for j in range(1,n+1)) for n in range(1,16)] # G. C. Greubel, Aug 29 2023

Formula

a(n) ~ c * n^(n^3/3 + n^2/4 - 7*n/12 + 5/8) * (2*Pi)^(n*(n-1)/4) / exp(4*n^3/9 - n^2/8 - n), where c = 0.488888619502150098591650327163991582267254151817880403495924251381414248582... - Vaclav Kotesovec, Nov 20 2023

Extensions

Name edited by Alois P. Heinz, Jul 23 2017

A325052 a(n) = Product_{i=0..n, j=0..n, k=0..n} (i! + j! + k!).

Original entry on oeis.org

3, 6561, 10319560704000000, 47749397192482757629144508002855841842593792000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Mar 26 2019

Keywords

Comments

Next term is too long to be included.

Crossrefs

Cf. A306594, A306729, A324425, A325053.

Programs

  • Mathematica
    Table[Product[i! + j! + k!, {i, 0, n}, {j, 0, n}, {k, 0, n}], {n, 0, 5}]
Clear[a]; a[n_] := a[n] = If[n == 0, 3, a[n-1] * Product[k! + j! + n!, {j, 0, n}, {k, 0, n}]^3 * (3*n!) / (Product[k! + 2*n!, {k, 0, n}]^3)]; Table[a[n], {n, 0, 5}]

Formula

a(n) ~ c * 2^(n^3/2 + 3*n^2 + 3*n) * 3^n * Pi^(n^3/2 + 3*n^2/2 + 3*n/2) * n^(3*n^4/4 + 3*n^3 + 17*n^2/4 + 5*n/2 + 601/120) / exp(15*n^4/16 + 3*n^3 + 3*n^2 - 21*n/4), where c = 28023.0953536911860317693532637428153075420958129597133...

A325048 a(n) = Product_{i=0..n, j=0..n} (i!^2 + j!^2).

Original entry on oeis.org

2, 16, 80000, 17272267776000000, 277884245560378426290863196025651200000000, 3337940951837185557810120427617693521487357301121536848574225250643001642844160000000000
Offset: 0

Views

Author

Vaclav Kotesovec, Mar 26 2019

Keywords

Crossrefs

Cf. A306729, A324403, A325050.

Programs

  • Mathematica
    Table[Product[i!^2 + j!^2, {i, 0, n}, {j, 0, n}], {n, 0, 7}]
Clear[a]; a[n_] := a[n] = If[n == 0, 2, a[n-1] * Product[k!^2 + n!^2, {k, 0, n}]^2 / (2*n!^2)]; Table[a[n], {n, 0, 7}]
  • Python
    from math import prod, factorial as f
def a(n): return prod(f(i)**2+f(j)**2 for i in range(n) for j in range(n))
print([a(n) for n in range(1, 7)]) # Michael S. Branicky, Feb 16 2021

Formula

a(n) ~ c * 2^(n*(n+3)) * Pi^(n*(n+2)) * n^((n+1)*(2*n+1)*(2*n+3)/3) / exp(2*n*(2*n+3)*(4*n+3)/9), where c = 401.488675138779168689540247334821476110398137334270208637438...

A325053 a(n) = Product_{i=0..n, j=0..n} (i! + j! + 1).

Original entry on oeis.org

3, 81, 103680, 447180963840, 7014935716261432173527040, 1921470539412808834455592518302690305036517376000, 81601182941928855942156180258180656419177691149082352022004942698629910149621350400000
Offset: 0

Views

Author

Vaclav Kotesovec, Mar 26 2019

Keywords

Crossrefs

Cf. A107254, A306729, A325052.

Programs

  • Mathematica
    Table[Product[i! + j! + 1, {i, 0, n}, {j, 0, n}], {n, 0, 7}]

Formula

a(n) = A306729(n) * Product_{i=0..n, j=0..n} (1 + 1/(i! + j!)).
a(n) ~ c * A324569 * 2^(n^2/2 + 2*n) * Pi^(n^2/2 + n) * n^(2*n^3/3 + 2*n^2 + 11*n/6 + 5/2) / exp(8*n^3/9 + 2*n^2 + n), where c = Product_{i>=0, j>=0} (1 + 1/(i! + j!)) = 71.32069635593350979104242285703294604508330622582076432053456223608...
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.