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.

A203519 a(n) = v(n+1)/v(n), where v=A203518.

Original entry on oeis.org

3, 20, 336, 12870, 1270080, 311323584, 197399802600, 321880885724160, 1365311591573529600, 15068868587132753685600, 434169705562891299584593920, 32678748925653999616045678080000, 6431834564578466234122576826339121600
Offset: 1

Views

Author

Clark Kimberling, Jan 03 2012

Keywords

Crossrefs

Cf. A203518, A000045, A093883.

Programs

  • Mathematica
    f[j_] := Fibonacci[j + 1]; z = 15;
v[n_] := Product[Product[f[k] + f[j], {j, 1, k - 1}], {k, 2, n}]
d[n_] := Product[(i - 1)!, {i, 1, n}] (* A000178(n-1) *)
Table[v[n], {n, 1, z}]                (* A203518 *)
Table[v[n + 1]/v[n], {n, 1, z - 1}]   (* A203519 *)
Table[v[n]/d[n], {n, 1, 20}]          (* A203520 *)

Formula

a(n) ~ c * phi^(n*(n+2) + 5/6) / 5^(n/2 + 1/4), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio and c = 3.990771264633156107481636998828550132941483550455485713064916076346986459357... - Vaclav Kotesovec, Apr 09 2021

A203520 v(n)/A000178(n); v=A203518 and A000178=(superfactorials).

Original entry on oeis.org

1, 3, 30, 1680, 900900, 9535125600, 4122929827336320, 161481256755920962660800, 1289130207153926967849156327590400, 4850265693548396005370498087328884780717568000, 20141097979706537636828034511787661382412368790843921121216000
Offset: 1

Views

Author

Clark Kimberling, Jan 03 2012

Keywords

Comments

It is conjectured that every term of A203520 is an integer.

Crossrefs

Cf. A203518, A000045, A000178, A093883.

Programs

  • Mathematica
    f[j_] := Fibonacci[j + 1]; z = 15;
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}]                (* A203518 *)
Table[v[n + 1]/v[n], {n, 1, z - 1}]   (* A203519 *)
Table[v[n]/d[n], {n, 1, 20}]          (* A203520 *)

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

A203311 Vandermonde determinant of (1,2,3,...,F(n+1)), where F=A000045 (Fibonacci numbers).

Original entry on oeis.org

1, 1, 1, 2, 48, 30240, 1596672000, 18172937502720000, 122457316443772566896640000, 1284319496829094129116119090331648000000, 55603466527142141932748234118927499493985767915520000000, 26110840958525805673462196263372614726154694067746586937781385166848000000000
Offset: 0

Views

Author

Clark Kimberling, Jan 01 2012

Keywords

Comments

Each term divides its successor, as in A123741. Each term is divisible by the corresponding superfactorial, A000178(n), as in A203313.
For a signed version, see A123742. For a guide to related sequences, including sequences of Vandermonde permanents, see A093883.

Examples

			v(4) = (2-1)*(3-1)*(3-2)*(5-1)*(5-2)*(5-3).

Crossrefs

Cf. A000045, A123741, A123742, A203313, A203518.

Programs

  • Maple
    with(LinearAlgebra): F:= combinat[fibonacci]:
a:= n-> Determinant(VandermondeMatrix([F(i)$i=2..n+1])):
seq(a(n), n=0..12);  # Alois P. Heinz, Jul 23 2017
  • Mathematica
    f[j_] := Fibonacci[j + 1]; z = 15;
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}]                (* A203311 *)
Table[v[n + 1]/v[n], {n, 1, z - 1}]   (* A123741 *)
Table[v[n]/d[n], {n, 1, 13}]          (* A203313 *)
  • Python
    from sympy import fibonacci, factorial
from operator import mul
from functools import reduce
def f(j): return fibonacci(j + 1)
def v(n): return 1 if n==1 else reduce(mul, [reduce(mul, [f(k) - f(j) for j in range(1, k)]) for k in range(2, n + 1)])
print([v(n) for n in range(1, 16)]) # Indranil Ghosh, Jul 26 2017

Formula

a(n) ~ c * d^n * phi^(n^3/3 + n^2/2) / 5^(n^2/4), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio, d = 0.120965069090607877853843907542896935455225485213927649233956250456604334... and c = 197.96410442333389877538426269... - Vaclav Kotesovec, Apr 08 2021
Showing 1-4 of 4 results.

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