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 1-4 of 4 results.

A203417 a(n) = A203415(n)/A000178(n).

Original entry on oeis.org

1, 3, 15, 140, 700, 2520, 44352, 2196480, 47567520, 634233600, 51753461760, 13984935444480, 1448751906201600, 82605199597240320, 32797812715211980800, 5296846753506734899200, 483765735240908144640000, 28985693293514522492928000
Offset: 1

Views

Author

Clark Kimberling, Jan 01 2012

Keywords

Links

Crossrefs

Cf. A000040, A018252, A203415, A203416.

Programs

  • Magma
    A018252:=[n : n in [1..250] | not IsPrime(n) ];
BarnesG:= func< n | (&*[Factorial(k): k in [0..n-2]]) >;
v:= func< n | n eq 1 select 1 else (&*[(&*[A018252[k+2] - A018252[j+1]: j in [0..k]]): k in [0..n-2]]) >;
[v(n)/BarnesG(n+1): n in [1..30]]; // G. C. Greubel, Feb 29 2024
  • Mathematica
    z=20;
nonprime = Join[{1}, Select[Range[250], CompositeQ]]; (* A018252 *)
f[j_]:= nonprime[[j]];
v[n_]:= Product[Product[f[k] - f[j], {j,1,k-1}], {k,2,n}];
d[n_]:= Product[(i-1)!, {i,1,n}];
Table[v[n], {n,1,z}]             (* A203415 *)
Table[v[n + 1]/v[n], {n,1,z}]    (* A203416 *)
Table[v[n]/d[n], {n,1,z}]        (* this sequence *)
  • SageMath
    A018252=[n for n in (1..250) if not is_prime(n)]
def BarnesG(n): return product(factorial(j) for j in range(1, n-1))
def v(n): return product(product(A018252[k-1]-A018252[j-1] for j in range(1,k)) for k in range(2,n+1))
[v(n)/BarnesG(n+1) for n in range(1,31)] # G. C. Greubel, Feb 29 2024

A203416 a(n) = A203415(n+1)/A203415(n).

Original entry on oeis.org

3, 10, 56, 120, 432, 12672, 249600, 873180, 4838400, 296110080, 10786406400, 49621572000, 355053404160, 34613526528000, 211189410432000, 1910897049600000, 21311651380219200, 274774815041126400, 62908970812047360000
Offset: 1

Views

Author

Clark Kimberling, Jan 01 2012

Keywords

Links

Crossrefs

Cf. A000040, A018252, A203415, A203417.

Programs

  • Magma
    A018252:=[n : n in [1..250] | not IsPrime(n) ];
A203416:= func< n | n eq 1 select 3 else (&*[A018252[n+1] - A018252[j+1]: j in [0..n-1]]) >;
[A203416(n): n in [1..30]]; // G. C. Greubel, Feb 29 2024
  • Mathematica
    z=20;
nonprime = Join[{1}, Select[Range[250], CompositeQ]]; (* A018252 *)
f[j_]:= nonprime[[j]];
v[n_]:= Product[Product[f[k] - f[j], {j,1,k-1}], {k,2,n}];
d[n_]:= Product[(i-1)!, {i,1,n}];
Table[v[n], {n,1,z}]             (* A203415 *)
Table[v[n+1]/v[n], {n,1,z}]      (* this sequence *)
Table[v[n]/d[n], {n,1,z}]        (* A203417 *)
  • SageMath
    A018252=[n for n in (1..250) if not is_prime(n)]
def A203416(n): return product(A018252[n]-A018252[j] for j in range(n))
[A203416(n) for n in range(1,31)] # G. C. Greubel, Feb 29 2024

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

A203527 a(n) = Product_{1 <= i < j <= n} (A018252(i) + A018252(j)); A018252 = nonprime numbers.

Original entry on oeis.org

1, 5, 350, 529200, 17542980000, 14783258730240000, 511420331138811494400000, 871980665589501641034301440000000, 60150685659205753788492548338089984000000000, 182771197941564481989784945231570147139911680000000000000
Offset: 1

Views

Author

Clark Kimberling, Jan 03 2012

Keywords

Comments

Each term divides its successor, as in A203528. It is conjectured that each term is divisible by the corresponding superfactorial, A000178(n); as in A203529. See A093883 for a guide to related sequences.

Crossrefs

Cf. A018252, A203415, A203528, A203529.

Programs

  • Maple
    b:= proc(n) option remember; local k; if n=1 then 1
      else for k from 1+b(n-1) while isprime(k) do od; k fi
    end:
a:= n-> mul(mul(b(i)+b(j), i=1..j-1), j=2..n):
seq(a(n), n=1..10);  # Alois P. Heinz, Jul 23 2017
  • Mathematica
    t = Table[If[PrimeQ[k], 0, k], {k, 1, 100}];
nonprime = Rest[Union[t]]              (* A018252 *)
f[j_] := nonprime[[j]]; z = 20;
v[n_] := Product[Product[f[k] + f[j], {j, 1, k - 1}], {k, 2, n}]
d[n_] := Product[(i - 1)!, {i, 1, n}]  (* A000178 *)
Table[v[n], {n, 1, z}]                 (* A203527 *)
Table[v[n + 1]/v[n], {n, 1, z - 1}]    (* A203528 *)
Table[v[n]/d[n], {n, 1, 20}]           (* A203529 *)

Extensions

Name edited by Alois P. Heinz, Jul 23 2017
Showing 1-4 of 4 results.

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