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 1-3 of 3 results.

A203434 a(n) = A203433(n)/A000178(n) where A000178=(superfactorials).

Original entry on oeis.org

1, 1, 3, 6, 45, 189, 3402, 30618, 1299078, 25332021, 2507870079, 106698472452, 24487299427734, 2283997201168644, 1209640056157393380, 248218139523497121576, 302358334494179897593596, 136861610819571430116630660
Offset: 1

Views

Author

Clark Kimberling, Jan 02 2012

Keywords

Links

Crossrefs

Cf. A000178, A007494, A014402, A203432, A203433.

Programs

  • Magma
    Barnes:= func< n | (&*[Factorial(j): j in [1..n-1]]) >;
f:= func< k | (&*[k+1-j+Floor((k+2)/2)-Floor((j+1)/2): j in [1..k]]) >;
[1] cat [(&*[f(k): k in [1..n-1]])/Barnes(n): n in [2..20]]; // G. C. Greubel, Sep 19 2023
  • Mathematica
    f[j_]:= j + Floor[(j+1)/2]; z = 20;
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}]             (* A203433 *)
Table[v[n+1]/v[n], {n,z}]      (* A014402 *)
Table[v[n]/d[n], {n,z}]        (* A203434 *)
  • SageMath
    def barnes(n): return product(factorial(j) for j in range(n))
def f(k): return product(k-j+(k//2)-(j//2) for j in range(k))
[product(f(k) for k in range(1, n) )//barnes(n) for n in range(1,31)] # G. C. Greubel, Sep 19 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

A014402 Numbers found in denominators of expansion of Airy function Ai(x).

Original entry on oeis.org

1, 1, 6, 12, 180, 504, 12960, 45360, 1710720, 7076160, 359251200, 1698278400, 109930867200, 580811212800, 46170964224000, 268334780313600, 25486372251648000, 161000868188160000, 17891433320656896000, 121716656350248960000, 15565546988971499520000, 113196490405731532800000
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Although the description is technically correct, this sequence is unsatisfactory because there are gaps in the series.
A014402 arises via Vandermonde determinants as in A203433; see the Mathematica section. - Clark Kimberling, Jan 02 2012

Examples

			Mathematica gives the series as 1/(3^(2/3)*Gamma(2/3)) - x/(3^(1/3)*Gamma(1/3)) + x^3/(6*3^(2/3)*Gamma(2/3)) - x^4/(12*3^(1/3)*Gamma(1/3)) + x^6/(180*3^(2/3)*Gamma(2/3)) - x^7/(504*3^(1/3)*Gamma(1/3)) + x^9/(12960*3^(2/3)*Gamma(2/3)) - ...

Links

Crossrefs

Cf. A014403, A060507, A176730, A176731, A203433, A203434.

Programs

  • Magma
    A014402:= func< n | n eq 0 select 1 else (&*[n-j+Floor(n/2)-Floor(j/2): j in [0..n-1]]) >;
[A014402(n): n in [0..25]]; // G. C. Greubel, Sep 20 2023
  • Mathematica
    Series[ AiryAi[ x ], {x, 0, 30} ]
a[ n_] := If[ n<0, 0, (n + Quotient[ n, 2])! / Product[ 3 k + 1 + Mod[n, 2], {k, 0, Quotient[ n, 2] - 1}]]; (* Michael Somos, Oct 14 2011 *)
(* Next, A014402 generated in via Vandermonde determinants based on A007494 *)
f[j_]:= j + Floor[(j+1)/2]; z = 20;
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}]             (* A203433 *)
Table[v[n+1]/v[n], {n,z}]      (* this sequence *)
Table[v[n]/d[n], {n,z}]        (* A203434 *)
(* Clark Kimberling, Jan 02 2012 *)
  • PARI
    {a(n) = if( n<0, 0, (n\2 + n)! / prod( k=0, n\2 -1, n%2 + 3*k + 1))}; /* Michael Somos, Oct 14 2011 */
  • SageMath
    def A014402(n): return product(n-j+(n//2)-(j//2) for j in range(n))
[A014402(n) for n in range(31)] # G. C. Greubel, Sep 20 2023

Formula

a(2*n) = A176730(n). a(2*n + 1) = A176731(n). - Michael Somos, Oct 14 2011
Showing 1-3 of 3 results.

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

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