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.

A203425 a(n) = w(n+1)/(4*w(n)), where w = A203424.

Original entry on oeis.org

-1, 9, -128, 2500, -62208, 1882384, -67108864, 2754990144, -128000000000, 6639980697856, -380420285792256, 23857239165420544, -1625527855624486912, 119574225000000000000, -9444732965739290427392
Offset: 1

Views

Author

Clark Kimberling, Jan 02 2012

Keywords

Links

Crossrefs

Cf. A203424.

Programs

  • Magma
    [(-2*(n+1))^n/4: n in [1..20]]; // G. C. Greubel, Dec 06 2023
  • Mathematica
    (* First program *)
f[j_] := 1/(2 j); z = 16;
v[n_] := Product[Product[f[k] - f[j], {j, 1, k - 1}], {k, 2, n}]
1/Table[v[n], {n, z}]              (* A203424 *)
Table[v[n]/(4 v[n + 1]), {n, z}]   (* A203425 *)
(* Second program *)
Table[(-2*(n+1))^n/4, {n, 20}] (* G. C. Greubel, Dec 06 2023 *)
  • PARI
    for(n=1, 25, print1((1/4)*(-2*(n+1))^n, ", ")) \\ G. C. Greubel, Jan 28 2017
  • SageMath
    [(-2*(n+1))^n/4 for n in range(1,21)] # G. C. Greubel, Dec 06 2023

Formula

a(n) = (1/4)*(-2*(n+1))^n. - Andrei Asinowski, Nov 03 2015
E.g.f.: (1/4)*(LambertW(2*x)/(2*x*(1 + LambertW(2*x))) - 1). - G. C. Greubel, Dec 06 2023

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

A203421 Reciprocal of Vandermonde determinant of (1,1/2,...,1/n).

Original entry on oeis.org

1, 1, -2, -18, 1152, 720000, -5598720000, -658683809280000, 1381360067999170560000, 59463021447701323327733760000, -59463021447701323327733760000000000000, -1542317635347398938581016812202229760000000000000
Offset: 0

Views

Author

Clark Kimberling, Jan 02 2012

Keywords

Comments

Each term divides its successor, as in A000169.

Links

Crossrefs

Cf. A000169, A002109, A057077, A203422, A203424.

Programs

  • Magma
    BarnesG:= func< n | (&*[Factorial(k): k in [0..n-2]]) >;
A203421:= func< n | (-1)^Binomial(n,2)*(Factorial(n))^n/BarnesG(n+2) >;
[A203421(n): n in [1..20]]; // G. C. Greubel, Dec 07 2023
  • Mathematica
    (* First program *)
f[j_] := 1/j; z = 12;
v[n_] := Product[Product[f[k] - f[j], {j, 1, k - 1}], {k, 2, n}]
Table[v[n], {n, 1, z}]
1/%  (* A203421 *)
Table[v[n]/v[n + 1], {n, 1, z}]  (* A000169 signed *)
(* Additional programs *)
Table[(-1)^Floor[n/2]*Product[(k + 1)^k, {k, 0, n-1}], {n, 1, 10}] (* Vaclav Kotesovec, Oct 18 2015 *)
Table[(-1)^Binomial[n,2]*(n!)^n/BarnesG[n+2], {n, 20}] (* G. C. Greubel, Dec 07 2023 *)
  • PARI
    a(n) = prod(i=2,n, (-i)^(i-1)); \\ Kevin Ryde, Apr 17 2022
  • SageMath
    def BarnesG(n): return product(factorial(k) for k in range(n-1))
def A203421(n): return (-1)^binomial(n, 2)*(gamma(n+1))^n/BarnesG(n+2)
[A203421(n) for n in range(1, 21)] # G. C. Greubel, Dec 07 2023

Formula

G.f.: G(0)/(2*x) -1/x, where G(k)= 1 + 1/(1 - x/(x + (2*k+1)/((2*k+1)^(2*k+1))/(1 + 1/(1 - x/(x - (2*k+2)/((2*k+2)^(2*k+2))/G(k+1)))))); (continued fraction). - Sergei N. Gladkovskii, Jun 03 2013
a(n) = (-1)^floor(n/2) * hyperfactorial(n)/n! = A057077(n) * A002109(n)/n!. - Paul J. Harvey, Feb 08 2014
a(n) = Product_{i=2..n} (-i)^(i-1). - Kevin Ryde, Apr 17 2022
abs(a(n)) ~ A * n^(n*(n-1)/2 - 5/12) / (sqrt(2*Pi) * exp(n^2/4 - n)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Nov 20 2023
a(n) = (-1)^binomial(n,2) * (n!)^n / BarnesG(n+2). - G. C. Greubel, Dec 07 2023

Extensions

a(0)=1 prepended by Alois P. Heinz, Apr 13 2024

A203426 Reciprocal of Vandermonde determinant of (1/4,1/6,...,1/(2n+2)).

Original entry on oeis.org

1, -12, -2304, 9216000, 955514880000, -3083393008926720000, -362115253665574567280640000, 1773553697494609431031516590243840000, 408626771902758012909661422392180736000000000000, -4933225232839126697329071833709661506078108549120000000000000
Offset: 1

Views

Author

Clark Kimberling, Jan 02 2012

Keywords

Comments

Each term divides its successor, as in A203427.

Links

Crossrefs

Cf. A203421, A203424, A203427.

Programs

  • Magma
    BarnesG:= func< n | (&*[Factorial(k): k in [0..n-2]]) >;
A203426:= func< n | (-2)^Binomial(n,2)*Factorial(n)*(Factorial(n+1))^n/BarnesG(n+3) >;
[A203426(n): n in [1..20]]; // G. C. Greubel, Dec 05 2023
  • Maple
    with(LinearAlgebra):
a:= n-> 1/Determinant(VandermondeMatrix([1/(2*i+2)$i=1..n])):
seq(a(n), n=1..12);  # Alois P. Heinz, Jul 23 2017
  • Mathematica
    (* First program *)
f[j_] := 1/(2 j + 2); z = 12;
v[n_] := Product[Product[f[k] - f[j], {j, 1, k - 1}], {k, 2, n}];
1/Table[v[n], {n, 1, z}]              (* A203426 *)
Table[v[n]/(4 v[n + 1]), {n, 1, z}]   (* A203427 *)
(* Second program *)
Table[(-2)^Binomial[n,2]*n!*(Gamma[n+2])^n/BarnesG[n+3], {n,20}] (* G. C. Greubel, Dec 05 2023 *)
  • SageMath
    def BarnesG(n): return product(factorial(k) for k in range(n-1))
def A203426(n): return (-2)^binomial(n,2)*gamma(n+1)*(gamma(n+2))^n/BarnesG(n+3)
[A203426(n) for n in range(1,21)] # G. C. Greubel, Dec 05 2023

Formula

a(n) = Product_{k=1..n} k * (-2(k+1))^(k-1). - Andrei Asinowski, Nov 03 2015
a(n) ~ (-1)^(n*(n-1)/2) * A * 2^(n^2/2 - n/2 - 1/2) * n^(n^2/2 + n/2 - 17/12) / (sqrt(Pi) * exp(n^2/4 - n - 1)), where A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Dec 05 2015
a(n) = (-2)^binomial(n,2) * n! * (Gamma(n+2))^n / BarnesG(n+3). - G. C. Greubel, Dec 05 2023

A203428 Reciprocal of Vandermonde determinant of (1/3,1/6,...,1/(3n)).

Original entry on oeis.org

1, -6, -486, 839808, 42515280000, -80335512599040000, -6890065294166289123840000, 31601087581187838970614157148160000, 8925080517850366815864624583251321642024960000
Offset: 1

Views

Author

Clark Kimberling, Jan 02 2012

Keywords

Comments

Each term divides its successor, as in A203429.

Links

Crossrefs

Cf. A203421, A203424, A203429.

Programs

  • Magma
    Barnes:= func< n | (&*[Factorial(j): j in [1..n-1]]) >;
A203428:= func< n | (-3)^Binomial(n,2)*(Factorial(n))^n/Barnes(n+1) >;
[A203428(n): n in [1..25]]; // G. C. Greubel, Sep 28 2023
  • Mathematica
    (* First program *)
f[j_]:= 1/(3*j); z = 16;
v[n_]:= Product[Product[f[k] - f[j], {j,k-1}], {k,2,n}]
1/Table[v[n], {n,z}]             (* A203428 *)
Table[v[n]/(3*v[n+1]), {n,z}]    (* A203429 *)
(* Second program *)
Table[(-3)^Binomial[n,2]*(Gamma[n+1])^(n-1)/BarnesG[n+1], {n,20}] (* G. C. Greubel, Sep 28 2023 *)
  • SageMath
    def barnes(n): return product(factorial(j) for j in range(n))
def A203428(n): return (-3)^binomial(n,2)*(factorial(n))^n/barnes(n+1)
[A203428(n) for n in range(1,21)] # G. C. Greubel, Sep 28 2023

Formula

a(n) = (-3)^binomial(n,2) * (Gamma(n+1))^(n-1) / BarnesG(n+1). - G. C. Greubel, Sep 28 2023
a(n) ~ (-1)^(n*(n-1)/2) * A * 3^(n*(n-1)/2) * n^(n*(n-1)/2 - 5/12) / (sqrt(2*Pi) * exp(n^2/4 - n)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Aug 09 2025
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