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%I #46 Nov 21 2023 04:33:06
%S 1,3,60,12600,38102400,2112397056000,2609908810629120000,
%T 84645606509847871488000000,82967862872337478796810649600000000,
%U 2781259372192376861719959017613164544000000000
%N Product of all possible sums of two distinct numbers taken from among first n natural numbers.
%C From _Clark Kimberling_, Jan 02 2013: (Start)
%C Each term divides its successor, as in A006963, and by the corresponding superfactorial, A000178(n), as in A203469.
%C 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).
%C Guide to related sequences:
%C ...
%C s(n).............. D(n)....... P(n)
%C n................. A000178.... (this)
%C n+1............... A000178.... A203470
%C n+2............... A000178.... A203472
%C n^2............... A202768.... A203475
%C 2^(n-1)........... A203303.... A203477
%C 2^n-1............. A203305.... A203479
%C n!................ A203306.... A203482
%C n(n+1)/2.......... A203309.... A203511
%C Fibonacci(n+1).... A203311.... A203518
%C prime(n).......... A080358.... A203521
%C odd prime(n)...... A203315.... A203524
%C nonprime(n)....... A203415.... A203527
%C composite(n)...... A203418.... A203530
%C 2n-1.............. A108400.... A203516
%C n+floor(n/2)...... A203430
%C n+floor[(n+1)/2].. A203433
%C 1/n............... A203421
%C 1/(n+1)........... A203422
%C 1/(2n)............ A203424
%C 1/(2n+2).......... A203426
%C 1/(3n)............ A203428
%C 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.
%C ...
%C 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:
%C ...
%C s(n) ............ x^2+xy+y^2.. x^2-xy+y^2.. x^2+y^2
%C n ............... A203012..... A203312..... A203475
%C n+1 ............. A203581..... A203583..... A203585
%C 2n-1 ............ A203514..... A203587..... A203589
%C n^2 ............. A203673..... A203675..... A203677
%C 2^(n-1) ......... A203679..... A203681..... A203683
%C n! .............. A203685..... A203687..... A203689
%C n(n+1)/2 ........ A203691..... A203693..... A203695
%C Fibonacci(n) .... A203742..... A203744..... A203746
%C Fibonacci(n+1) .. A203697..... A203699..... A203701
%C prime(n) ........ A203703..... A203705..... A203707
%C floor(n/2) ...... A203748..... A203752..... A203773
%C floor((n+1)/2) .. A203759..... A203763..... A203766
%C For f(x,y)=x^4+y^4, see A203755 and A203770. (End)
%D Amarnath Murthy, Another combinatorial approach towards generalizing the AM-GM inequality, Octagon Mathematical Magazine, Vol. 8, No. 2, October 2000.
%D 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.
%H T. D. Noe, <a href="/A093883/b093883.txt">Table of n, a(n) for n = 1..20</a>
%F Partial products of A006963: a(n) = Product((2*i-1)!/i!, i=1..n). - _Vladeta Jovovic_, May 27 2004
%F 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
%F 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
%e a(4) = (1+2)*(1+3)*(1+4)*(2+3)*(2+4)*(3+4) = 12600.
%p a:= n-> mul(mul(i+j, i=1..j-1), j=2..n):
%p seq(a(n), n=1..12); # _Alois P. Heinz_, Jul 23 2017
%t f[n_] := Product[(j + k), {k, 2, n}, {j, 1, k - 1}]; Array[f, 10] (* _Robert G. Wilson v_, Jan 08 2013 *)
%o (PARI) A093883(n)=prod(i=1,n,(2*i-1)!/i!) \\ _M. F. Hasler_, Nov 02 2012
%Y Cf. A006963, A093884, A203469.
%K nonn
%O 1,2
%A _Amarnath Murthy_, Apr 22 2004
%E More terms from _Vladeta Jovovic_, May 27 2004
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