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 1-6 of 6 results.

A203513 a(n) = A203312(n+1)/A203312(n).

Original entry on oeis.org

3, 49, 2028, 159201, 20342448, 3850078401, 1012487793408, 353293863908769, 157973407966483200, 88087149666575064369, 59928191584204259377152, 48860028872008706126041281
Offset: 1

Views

Author

Clark Kimberling, Jan 04 2012

Keywords

Comments

See A093883 for a discussion and guide to related sequences.

Links

Crossrefs

Cf. A093883, A203312.

Programs

  • Magma
    [(&*[(n+1)*(n-j+1) +j^2: j in [1..n]]): n in [1..30]]; // G. C. Greubel, Feb 23 2024
  • Mathematica
    f[j_] := j; z = 12;
v[n_] := Product[Product[f[j]^2 - f[j] f[k] + f[k]^2,
{j, 1, k - 1}], {k, 2, n}]
Table[v[n], {n, 1, z}]          (* A203312 *)
Table[v[n + 1]/v[n], {n, 1, z}] (* A203513 *)
Table[Product[k^2 - k*(n+1) + (n+1)^2, {k, 1, n}], {n, 1, 15}] (* Vaclav Kotesovec, Sep 07 2023 *)
  • Python
    from operator import mul
from functools import reduce
def v(n): return 1 if n==1 else reduce(mul, [reduce(mul, [j**2 - j*k + k**2 for j in range(1, k)]) for k in range(2, n + 1)])
print([v(n + 1)//v(n) for n in range(1, 13)]) # Indranil Ghosh, Jul 26 2017
  • SageMath
    def A203513(n): return product((n+1)*(n-j+1) +j^2 for j in range(1, n+1))
[A203513(n) for n in range(1,31)] # G. C. Greubel, Feb 23 2024

Formula

a(n) ~ exp((n+1)*Pi/(sqrt(3)) - 2*n) * n^(2*n). - Vaclav Kotesovec, Sep 07 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

A367543 a(n) = Product_{i=1..n, j=1..n} (i^2 - i*j + j^2).

Original entry on oeis.org

1, 36, 777924, 51190934086656, 32435802373365731229926400, 483207398728525904876601066508152707481600, 350969035472356907726779584093506665415605824531908346799718400
Offset: 1

Views

Author

Vaclav Kotesovec, Nov 22 2023

Keywords

Crossrefs

Cf. A203312, A324403, A367542.
Cf. A324426, A079478.

Programs

  • Mathematica
    Table[Product[Product[(i^2 - i*j + j^2), {i, 1, n}], {j, 1, n}], {n, 1, 10}]
  • Python
    from math import prod, factorial
def A367543(n): return (prod(i*(i-j)+j**2 for i in range(1,n) for j in range(i+1,n+1))*factorial(n))**2 # Chai Wah Wu, Nov 22 2023

Formula

a(n) = A324426(n) / A079478(n).
a(n) ~ 3^(1/6) * Gamma(1/3)^2 * n^(2*n^2 - 1/3) / (2^(5/3) * Pi^(5/3) * exp(3*n^2 - (n^2 + n + 1/6)*Pi/sqrt(3))).

A203673 Vandermonde sequence using x^2 + xy + y^2 applied to (1,4,9,...,n^2).

Original entry on oeis.org

1, 21, 254163, 11213968422384, 6451450005117349260375984, 127857993263632065817610313129228311433216, 191199773886534869435599958788731398661833328276349525268803584
Offset: 1

Views

Author

Clark Kimberling, Jan 04 2012

Keywords

Comments

See A093883 for a discussion and guide to related sequences.

Crossrefs

Cf. A367550.

Programs

  • Mathematica
    f[j_] := j^2; z = 12;
u[n_] := Product[f[j]^2 + f[j] f[k] + f[k]^2, {j, 1, k - 1}]
v[n_] := Product[u[n], {k, 2, n}]
Table[v[n], {n, 1, z}]          (* A203673 *)
Table[v[n + 1]/v[n], {n, 1, z}] (* A203674 *)

Formula

From Vaclav Kotesovec, Nov 22 2023: (Start)
a(n) = A203012(n) * A203312(n).
a(n) ~ c * 3^(n*(3*n+1)/4) * n^(2*n^2 - 2*n - 3/2) / exp(3*n^2 - n*(n+1)*Pi*sqrt(3)/4 - 2*n), where c = Gamma(1/3)^(3/2) * 3^(7/24) * exp(Pi/(8*sqrt(3))) / (2^(5/2) * Pi^(5/2)) = 0.076580853261060033865281896312127877504662138809362419847380161198324... (End)

A203012 Vandermonde sequence using x^2 + xy + y^2 applied to (1,2,...,n).

Original entry on oeis.org

1, 7, 1729, 37616124, 135933424914924, 132432199651531695045312, 51437933151214684812682944045953088, 11056394929890243558409721156996503083526683082752, 1743892714865607005898689849291524734866677095031979100765833773056
Offset: 1

Views

Author

Clark Kimberling, Jan 04 2012

Keywords

Comments

See A093883 for a discussion and guide to related sequences.

Examples

			a(1)=1
a(2)=1^2+1*2+2^2=7
a(3)=(1^2+1*2+2^2)(1^3+1*3+3^2)(2^2+2*3+3^2)=1729.

Crossrefs

Cf. A203312, A203475, A203673, A367542.

Programs

  • Mathematica
    f[j_] := j; z = 12;
v[n_] := Product[Product[f[j]^2 + f[j] f[k] + f[k]^2,
{j, 1, k - 1}], {k, 2, n}]
Table[v[n], {n, 1, z}]          (* A203012 *)
Table[v[n + 1]/v[n], {n, 1, z}] (* A203158 *)

Formula

a(n) ~ c * n^(n^2 - n - 5/6) * 3^(n*(3*n+1)/4) / exp(3*n^2/2 - n - n*(n+1)*Pi / (4*sqrt(3))), where c = sqrt(Gamma(1/3)) * 3^(5/24) * exp(Pi/(24*sqrt(3))) / (2^(7/6) * Pi^(7/6)) = 0.26001211479205772659823692637002123572622409280442625312217301129630097... - Vaclav Kotesovec, Nov 22 2023

A367517 a(n) = Product_{1 <= i < j <= n} (i^3 + j^3).

Original entry on oeis.org

1, 9, 8820, 3756261600, 1808353823416358400, 2039434890206782239939575808000, 9701283544768145414379038964290421034844160000, 318565016660642285381616434022976430918222575100233223503872000000
Offset: 1

Views

Author

Vaclav Kotesovec, Nov 21 2023

Keywords

Crossrefs

Cf. A203475, A324426.
Cf. A093883, A203312.

Programs

  • Mathematica
    Table[Product[Product[i^3 + j^3, {i,1,j-1}], {j,2,n}], {n,1,10}]
  • Python
    from math import prod
def A367517(n): return prod(i**3+j**3 for i in range(1,n) for j in range(i+1,n+1)) # Chai Wah Wu, Nov 22 2023

Formula

a(n) ~ c * 2^(n^2 + n/2) * n^(3*n*(n-1)/2 - 9/8) / exp(9*n^2/4 - Pi*n*(n+1) / (2*sqrt(3)) - 3*n/2), where c = 0.1474592465511237361695489773766942970580292170583524962797073759996256...
a(n) = A093883(n) * A203312(n).
Showing 1-6 of 6 results.

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