A203012 -id:A203012 - cp's OEIS frontend

A093883 Product of all possible sums of two distinct numbers taken from among first n natural numbers.

1, 3, 60, 12600, 38102400, 2112397056000, 2609908810629120000, 84645606509847871488000000, 82967862872337478796810649600000000, 2781259372192376861719959017613164544000000000

Offset: 1

Amarnath Murthy, Apr 22 2004

nonn

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)

			a(4) = (1+2)*(1+3)*(1+4)*(2+3)*(2+4)*(3+4) = 12600.

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.

T. D. Noe, Table of n, a(n) for n = 1..20

Cf. A006963, A093884, A203469.

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

f[n_] := Product[(j + k), {k, 2, n}, {j, 1, k - 1}]; Array[f, 10] (* Robert G. Wilson v, Jan 08 2013 *)

A093883(n)=prod(i=1,n,(2*i-1)!/i!)  \\ M. F. Hasler, Nov 02 2012

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

More terms from Vladeta Jovovic, May 27 2004

A367542 a(n) = Product_{i=1..n, j=1..n} (i^2 + i*j + j^2).

Original entry on oeis.org

3, 1764, 2905736652, 66016970246853190656, 64657853715047202043531429875379200, 6627957368676918780503749855130249245999452089509478400

Offset: 1

Vaclav Kotesovec, Nov 22 2023

nonn

Cf. A203012, A324403, A367543.

Table[Product[Product[(i^2 + i*j + j^2), {i, 1, n}], {j, 1, n}], {n, 1, 10}]

from math import prod, factorial
def A367542(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*3**n # Chai Wah Wu, Nov 22 2023

a(n) ~ c * 3^(3*n*(n+1)/2) * n^(2*n^2 - 2/3) / exp(3*n^2 - Pi*n*(n+1) / (2*sqrt(3))), where c = 3^(5/12) * exp(Pi/(12*sqrt(3))) * Gamma(1/3) / (2^(4/3) * Pi^(4/3)) = 0.42478290981890921418850643030484274341562970375995548434917...

A203312 Vandermonde sequence using x^2 - xy + y^2 applied to (1,2,...,n).

Original entry on oeis.org

1, 3, 147, 298116, 47460365316, 965460013501733568, 3717096745012192786213464768, 3763515081241454304168766426610670649344, 1329626784930718063722475681347135527472012731205697536

Offset: 1

Clark Kimberling, Jan 04 2012

nonn

See A093883 for a discussion and guide to related sequences.

Cf. A203012, A203673, A367543.

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 *)

from operator import mul
from functools import reduce
def v(n): return 1 if n==1 else reduce(mul, [j**2 - j*k + k**2 for k in range(2, n + 1) for j in range(1, k)])
print([v(n) for n in range(1, 11)]) # Indranil Ghosh, Jul 26 2017

a(n) ~ c * n^(n^2 - n - 2/3) / exp(3*n^2/2 - n*(n+1)*Pi/(2*sqrt(3)) - n), where c = Gamma(1/3) * 3^(1/12) * exp(Pi/(12*sqrt(3))) / (2^(4/3) * Pi^(4/3)) = 0.2945280196744096322469352538791946777977998766871923997662057483092872... - Vaclav Kotesovec, Nov 22 2023

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

Clark Kimberling, Jan 04 2012

nonn

See A093883 for a discussion and guide to related sequences.

Cf. A367550.

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 *)

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)

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A203158 v(n+1)/v(n), where v=A203012.

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A093883 Product of all possible sums of two distinct numbers taken from among first n natural numbers.

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A367542 a(n) = Product_{i=1..n, j=1..n} (i^2 + i*j + j^2).

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A203312 Vandermonde sequence using x^2 - xy + y^2 applied to (1,2,...,n).

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

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