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.

A203476 a(n) = v(n+1)/v(n), where v = A203475.

Original entry on oeis.org

5, 130, 8500, 1051076, 211255200, 62840245000, 25959932960000, 14224928867370000, 9986120745657472000, 8740787543400204500000, 9333385482079885824000000, 11942338721669302523305000000, 18038821394494464638896640000000
Offset: 1

Views

Author

Clark Kimberling, Jan 02 2012

Keywords

Links

Crossrefs

Cf. A093883, A110468, A203475.

Programs

  • Magma
    [(&*[(n+1)^2 + j^2: j in [1..n]]): n in [1..20]]; // G. C. Greubel, Aug 28 2023
  • Mathematica
    (* First program *)
f[j_]:= j^2; z = 15;
v[n_]:= Product[Product[f[k] + f[j], {j,k-1}], {k,2,n}]
Table[v[n], {n,z}]           (* A203475 *)
Table[v[n+1]/v[n], {n,z-1}]  (* A203476 *)
(* Second program *)
Table[Product[j^2 +(n+1)^2 , {j,n}], {n,20}] (* G. C. Greubel, Aug 28 2023 *)
  • SageMath
    [product(j^2+(n+1)^2 for j in range(1,n+1)) for n in range(1,21)] # G. C. Greubel, Aug 28 2023

Formula

a(n) ~ 2^(n + 1/2) * exp(Pi*(n+1)/2 - 2*n) * n^(2*n). - Vaclav Kotesovec, Jan 25 2019
a(n) = Product_{j=1..n} ((n+1)^2 + j^2). - G. C. Greubel, Aug 28 2023

A323755 Decimal expansion of a constant related to the asymptotics of A203475.

Original entry on oeis.org

2, 7, 4, 5, 2, 8, 3, 5, 0, 3, 3, 3, 5, 5, 2, 9, 0, 3, 8, 0, 0, 4, 0, 8, 9, 9, 3, 4, 8, 2, 5, 0, 7, 4, 2, 8, 1, 4, 2, 3, 8, 3, 7, 8, 3, 7, 7, 3, 1, 9, 0, 4, 5, 1, 1, 8, 1, 0, 7, 2, 7, 2, 3, 7, 4, 2, 6, 9, 1, 6, 7, 8, 1, 0, 5, 7, 6, 2, 7, 4, 0, 6, 2, 0, 0, 1, 6, 5, 7, 0, 3, 2, 1, 2, 1, 9, 6, 1, 2, 5, 7, 4, 4, 5, 2, 8
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 26 2019

Keywords

Examples

			0.274528350333552903800408993482507428142383783773190451181072723742691...

Links

Crossrefs

Cf. A068466, A203475.

Programs

  • Mathematica
    RealDigits[Sqrt[Gamma[1/4]] * E^(Pi/24) / (2^(9/8) * Pi^(9/8)), 10, 120][[1]]

Formula

Equals limit_{n->infinity} A203475(n) / (2^(n^2/2) * exp(Pi*n*(n+1)/4 - 3*n^2/2 + n) * n^(n*(n-1) - 3/4)).
Equals sqrt(Gamma(1/4)) * exp(Pi/24) / (2*Pi)^(9/8).

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

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

A293290 a(n) = Product_{1 <= j <= k <= n} (k^2 + j^2).

Original entry on oeis.org

1, 2, 80, 187200, 50918400000, 2675955409920000000, 40702283662588674048000000000, 250658664786823821917343252480000000000000, 832906513114759565863066815448211678822400000000000000000, 1919381816160714520414106848157314737202346840876384256000000000000000000000
Offset: 0

Views

Author

Velin Yanev, Oct 05 2017

Keywords

Crossrefs

Suggested by Omar E. Pol from A264596 formula.
Cf. A101686, A203475, A324403.

Programs

  • Mathematica
    Table[Product[k^2 + j^2, {k, 1, n}, {j, 1, k}], {n, 0, 10}]
  • Sage
    [prod([prod([k^2+j^2 for j in range(1,k+1)]) for k in range(1,n+1)]) for n in range(10)] # Danny Rorabaugh, Oct 16 2017

Formula

a(n) ~ sqrt(Gamma(1/4)) * Pi^(-1/8) * 2^(n^2/2 + n - 1/8) * exp(Pi*n*(n+1)/4 - 3*n^2/2 - n + Pi/24) * n^(n*(n+1) + 1/4). - Vaclav Kotesovec, Feb 26 2019

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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