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

1,2

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<k<=n}, which is the Vandermonde determinant if f(x,y)=y-x and the Vandermonde permanent if f(x,y)=x+y.

...

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)

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

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

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

Example

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

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 *)

Prog

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

Crossrefs

Cf. A006963, A093884, A203469.

Sequence in context: A165626 A120307 A022915 * A203518 A297562 A128075

Adjacent sequences: A093880 A093881 A093882 * A093884 A093885 A093886

Keyword

nonn

Author

Amarnath Murthy, Apr 22 2004

Extensions

More terms from Vladeta Jovovic, May 27 2004

Status

approved