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.

A203308 a(n) = A203306(n+1)/A203306(n).

Original entry on oeis.org

1, 1, 20, 9108, 153675648, 153926018668800, 13624548214772203315200, 148312029363286484759480524800000, 262925014428462931164318003384701335633920000, 96950311125839455466119755365478799838570665250861875200000
Offset: 0

Views

Author

Clark Kimberling, Jan 01 2012

Keywords

Links

Crossrefs

Cf. A203306, A323717.

Programs

  • Magma
    F:= Factorial; [1] cat [(&*[F(n+1) - F(j): j in [1..n]]): n in [1..20]]; // G. C. Greubel, Aug 30 2023
  • Mathematica
    (* First program *)
f[j_]:= j!; z = 10;
v[n_]:= Product[Product[f[k] - f[j], {j,k-1}], {k,2,n}]
Table[v[n], {n,0,z}]          (* A203306 *)
Table[v[n+1]/v[n], {n,0,z}]   (* A203308 *)
(* Second program *)
Table[Product[(n+1)! - k!, {k,n}], {n,0,10}] (* Vaclav Kotesovec, Jan 25 2019 *)
  • Python
    from sympy import factorial as f
from operator import mul
from functools import reduce
def v(n):
    return 1 if n<2 else reduce(mul, (f(k+1) - f(j) for k in range(1,n) for j in range(1, k+1)))
print([v(n + 1)//v(n) for n in range(16)]) # Indranil Ghosh, Jul 24 2017
  • SageMath
    f=factorial; [product(f(n+1) - f(k) for k in range(1,n+1)) for n in range(21)] # G. C. Greubel, Aug 30 2023

Formula

a(n) ~ (2*Pi)^(n/2) * n^(n*(2*n + 3)/2) / exp(n^2 - 13/12). - Vaclav Kotesovec, Jan 25 2019
a(n) = Product_{j=1..n} ((n+1)! - j!). - G. C. Greubel, Aug 30 2023

Extensions

a(0) = 1 prepended by G. C. Greubel, Aug 30 2023

A203466 a(n) = A203306(n)/A000178(n) where A000178 are superfactorials.

Original entry on oeis.org

1, 1, 10, 15180, 97199847360, 124679879327832253286400, 2359315315713931476611812172370616909824000, 69427548091550819116702789435220590352184299509517898727953530880000000
Offset: 1

Views

Author

Clark Kimberling, Jan 02 2012

Keywords

Links

Crossrefs

Cf. A203306.

Programs

  • Magma
    F:= Factorial; [1] cat [(&*[(&*[F(k+1) - F(j): j in [1..k]])/Factorial(k): k in [1..n-1]]): n in [2..20]]; // G. C. Greubel, Sep 19 2023
  • Mathematica
    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}]
Table[v[n], {n,z}]       (* A203306 *)
Table[v[n]/d[n], {n,z}]  (* A203466 *)
  • SageMath
    f=factorial; [product(product(f(k+1) - f(j) for j in range(1, k+1))//factorial(k) for k in range(1, n)) for n in range(1,21)] # G. C. Greubel, Sep 19 2023

Extensions

Name edited by Michel Marcus, May 17 2019

A323720 Decimal expansion of a constant related to the asymptotics of A203306.

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, Jan 25 2019

Keywords

Examples

			0.29363504888070220142364974947015983077985979143334289754405669958...

Crossrefs

Cf. A203306.

Formula

Equals limit_{n->infinity} A203306(n) / ((2*Pi)^(n*(n-1)/4) * n^(n^3/3 + n^2/4 - 7*n/12 - 11/8) / exp(4*n^3/9 - n^2/8 - n)).

A093883 Product of all possible sums of two distinct numbers taken from among first n natural numbers.

Original entry on oeis.org

1, 3, 60, 12600, 38102400, 2112397056000, 2609908810629120000, 84645606509847871488000000, 82967862872337478796810649600000000, 2781259372192376861719959017613164544000000000
Offset: 1

Views

Author

Amarnath Murthy, Apr 22 2004

Keywords

Comments

From Clark Kimberling, Jan 02 2013: (Start)
Each term divides its successor, as in A006963, and by the corresponding superfactorial, A000178(n), as in A203469.
Abbreviate "Vandermonde" as V. The V permanent of a set S={s(1),s(2),...,s(n)} is a product of sums s(j)+s(k) in analogy to the V determinant as a product of differences s(k)-s(j). Let D(n) and P(n) denote the V determinant and V permanent of S, and E(n) the V determinant of the numbers s(1)^2, s(2)^2, ..., s(n)^2; then P(n) = E(n)/D(n). This is one of many divisibility properties associated with V determinants and permanents. Another is that if S consists of distinct positive integers, then D(n) divides D(n+1) and P(n) divides P(n+1).
Guide to related sequences:
...
s(n).............. D(n)....... P(n)
n................. A000178.... (this)
n+1............... A000178.... A203470
n+2............... A000178.... A203472
n^2............... A202768.... A203475
2^(n-1)........... A203303.... A203477
2^n-1............. A203305.... A203479
n!................ A203306.... A203482
n(n+1)/2.......... A203309.... A203511
Fibonacci(n+1).... A203311.... A203518
prime(n).......... A080358.... A203521
odd prime(n)...... A203315.... A203524
nonprime(n)....... A203415.... A203527
composite(n)...... A203418.... A203530
2n-1.............. A108400.... A203516
n+floor(n/2)...... A203430
n+floor[(n+1)/2].. A203433
1/n............... A203421
1/(n+1)........... A203422
1/(2n)............ A203424
1/(2n+2).......... A203426
1/(3n)............ A203428
Generalizing, suppose that f(x,y) is a function of two variables and S=(s(1),s(2),...s(n)). The phrase, "Vandermonde sequence using f(x,y) applied to S" means the sequence a(n) whose n-th term is the product f(s(j,k)) : 1<=j
...
If f(x,y) is a (bivariate) cyclotomic polynomial and S is a strictly increasing sequence of positive integers, then a(n) consists of integers, each of which divides its successor. Guide to sequences for which f(x,y) is x^2+xy+y^2 or x^2-xy+y^2 or x^2+y^2:
...
s(n) ............ x^2+xy+y^2.. x^2-xy+y^2.. x^2+y^2
n ............... A203012..... A203312..... A203475
n+1 ............. A203581..... A203583..... A203585
2n-1 ............ A203514..... A203587..... A203589
n^2 ............. A203673..... A203675..... A203677
2^(n-1) ......... A203679..... A203681..... A203683
n! .............. A203685..... A203687..... A203689
n(n+1)/2 ........ A203691..... A203693..... A203695
Fibonacci(n) .... A203742..... A203744..... A203746
Fibonacci(n+1) .. A203697..... A203699..... A203701
prime(n) ........ A203703..... A203705..... A203707
floor(n/2) ...... A203748..... A203752..... A203773
floor((n+1)/2) .. A203759..... A203763..... A203766
For f(x,y)=x^4+y^4, see A203755 and A203770. (End)

Examples

			a(4) = (1+2)*(1+3)*(1+4)*(2+3)*(2+4)*(3+4) = 12600.

References

  • Amarnath Murthy, Another combinatorial approach towards generalizing the AM-GM inequality, Octagon Mathematical Magazine, Vol. 8, No. 2, October 2000.
  • Amarnath Murthy, Smarandache Dual Symmetric Functions And Corresponding Numbers Of The Type Of Stirling Numbers Of The First Kind. Smarandache Notions Journal, Vol. 11, No. 1-2-3 Spring 2000.

Links

Crossrefs

Cf. A006963, A093884, A203469.

Programs

  • Maple
    a:= n-> mul(mul(i+j, i=1..j-1), j=2..n):
seq(a(n), n=1..12);  # Alois P. Heinz, Jul 23 2017
  • Mathematica
    f[n_] := Product[(j + k), {k, 2, n}, {j, 1, k - 1}]; Array[f, 10] (* Robert G. Wilson v, Jan 08 2013 *)
  • PARI
    A093883(n)=prod(i=1,n,(2*i-1)!/i!)  \\ M. F. Hasler, Nov 02 2012

Formula

Partial products of A006963: a(n) = Product((2*i-1)!/i!, i=1..n). - Vladeta Jovovic, May 27 2004
G.f.: G(0)/(2*x) -1/x, where G(k)= 1 + 1/(1 - 1/(1 + 1/((2*k+1)!/(k+1)!)/x/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 15 2013
a(n) ~ sqrt(A/Pi) * 2^(n^2 + n/2 - 7/24) * exp(-3*n^2/4 + n/2 - 1/24) * n^(n^2/2 - n/2 - 11/24), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Jan 26 2019

Extensions

More terms from Vladeta Jovovic, May 27 2004

A203310 a(n) = A203309(n+1)/A203309(n).

Original entry on oeis.org

1, 2, 15, 252, 7560, 356400, 24324300, 2270268000, 277880803200, 43197833952000, 8315583035760000, 1942008468966720000, 540988073497872000000, 177227692877902867200000, 67457290601651778828000000, 29522484828017013792960000000, 14721879100904484211422720000000
Offset: 0

Views

Author

Clark Kimberling, Jan 01 2012

Keywords

Links

Crossrefs

Cf. A203306, A203309.

Programs

  • Magma
    F:= Factorial; [(F(n)*F(2*n+2))/(2^n*F(n+2)): n in [0..20]]; // G. C. Greubel, Aug 29 2023
  • Maple
    b:= proc(n) option remember; uses LinearAlgebra;
      Determinant(VandermondeMatrix([i*(i+1)/2$i=1..n]))
    end:
a:= n-> b(n+1)/b(n):
seq(a(n), n=0..16);  # Alois P. Heinz, Aug 29 2023
  • Mathematica
    (* First program *)
f[j_]:= j*(j+1)/2; z = 15;
v[n_]:= Product[Product[f[k] - f[j], {j,k-1}], {k,2,n}]
Table[v[n], {n,z}]             (* A203309 *)
Table[v[n+1]/v[n], {n,0,z-1}]  (* A203310 *)
(* Second program *)
Table[(n!*(2*n+2)!)/(2^n*(n+2)!), {n,0,20}] (* G. C. Greubel, Aug 29 2023 *)
  • Python
    from operator import mul
from functools import reduce
def f(n): return n*(n + 1)//2
def v(n): return 1 if n==1 else reduce(mul, (f(k) - f(j) for k in range(2, n + 1) for j in range(1, k)))
print([v(n + 1)//v(n) for n in range(1, 15)]) # Indranil Ghosh, Jul 24 2017
  • SageMath
    f=factorial; [(f(n)*f(2*n+2))/(2^n*f(n+2)) for n in range(21)] # G. C. Greubel, Aug 29 2023

Formula

a(n) ~ sqrt(Pi) * 2^(n+3) * n^(2*n + 1/2) / exp(2*n). - Vaclav Kotesovec, Jan 25 2019
a(n) = (n!*(2*n+2)!)/(2^n*(n+2)!). - G. C. Greubel, Aug 29 2023

Extensions

Name corrected by Vaclav Kotesovec, Jan 25 2019
a(0)=1 prepended by Alois P. Heinz, Aug 29 2023
Showing 1-5 of 5 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.