A203305 -id:A203305 - 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

A028365 Order of general affine group over GF(2), AGL(n,2).

Original entry on oeis.org

1, 2, 24, 1344, 322560, 319979520, 1290157424640, 20972799094947840, 1369104324918194995200, 358201502736997192984166400, 375234700595146883504949480652800, 1573079924978208093254925489963584716800

Offset: 0

N. J. A. Sloane

For n > 0, a(n) = v(n+1)/v(n), where v = A203305 is the Vandermonde determinant of the first n of the numbers -2^j - 1; see the Mathematica section. - Clark Kimberling, Jan 01 2012

J. M. Borwein, D. H. Bailey and R. Girgensohn, Experimentation in Mathematics, A K Peters, Ltd., Natick, MA, 2004. x+357 pp. See p. 54 (1.64).

Seiichi Manyama, Table of n, a(n) for n = 0..57
Abdalla G. M. Ahmed, Matt Pharr, Victor Ostromoukhov, and Hui Huang, SZ Sequences: Binary-Based (0, 2^q)-Sequences, arXiv:2505.20434 [cs.GR], 2025. See p. 7.
Marcus Brinkmann, Extended Affine and CCZ Equivalence up to Dimension 4, Ruhr University Bochum (2019).
Putnam Competition 1999, Question A6, Amer. Math. Monthly 107 (Oct 2000), 721-732; see p. 725.
I. Strazdins, Universal affine classification of Boolean functions, Acta Applic. Math. 46 (1997), 147-167.

Cf. A000217, A002884, A005329, A020522, A048651, A053601, A203305.

[1] cat [(&*[2^(n+1) - 2^(j+1): j in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Aug 31 2023

A028365 := n->2^n*product(2^n-2^'i','i'=0..n-1); # version 1
A028365 := n->product(2^'j'-1,'j'=1..n)*2^binomial(n+1,2); # version 2

RecurrenceTable[{a[1]==1, a[2]==2, a[3]==24, a[n]==(6a[n-1]^2 a[n-3] - 8a[n-1] a[n-2]^2)/(a[n-2] a[n-3])}, a[n], {n,20}] (* Harvey P. Dale, Aug 03 2011 *)
(* Next, the connection with Vandermonde determinants *)
f[j_]:= 2^j - 1; z = 15;
v[n_]:= Product[Product[f[k] - f[j], {j,k-1}], {k,2,n}]
Table[v[n], {n,z}]   (* A203303 *)
Table[v[n+1]/v[n], {n,z}]  (* A028365 *)
Table[v[n]*v[n+2]/(2*v[n+1])^2, {n,z}]  (* A171499 *) (* Clark Kimberling, Jan 01 2011 *)
Table[(-1)^n*2^Binomial[n+1,2]*QPochhammer[2,2,n], {n,0,20}] (* G. C. Greubel, Aug 31 2023 *)

a(n)=if(n<0,0,prod(k=1,n,2^k-1)*2^((n^2+n)/2)) /* Michael Somos, May 09 2005 */

[product(2^(n+1) - 2^(k+1) for k in range(n)) for n in range(21)] # G. C. Greubel, Aug 31 2023

a(n) is asymptotic to C*2^(n*(n+1)) where C = 0.288788095086602421278899721... = prod(k>=1, 1-1/2^k) (cf. A048651). - Benoit Cloitre, Apr 11 2003
a(n) = (6*a(n-1)^2*a(n-3) - 8*a(n-1)*a(n-2)^2) / (a(n-2)*a(n-3)). [From Putman Exam]. - Max Alekseyev, May 18 2007
a(n) = 2*A203305(n), n > 0. - Clark Kimberling, Jan 01 2012
From Max Alekseyev, Jun 09 2015: (Start)
a(n) = 2^A000217(n) * A005329(n).
a(n) = 2^n * A002884(n).
a(n) = 2^n * n! * A053601(n). (End)
From G. C. Greubel, Aug 31 2023: (Start)
a(n) = Product_{j=0..n-1} (2^(n+1) - 2^(j+1)).
a(n) = (-1)^n * 2^binomial(n+1,2) * QPochhammer(2,2,n). (End)

A203479 a(n) = Product_{1 <= i < j <= n} (2^i + 2^j - 2).

Original entry on oeis.org

1, 4, 320, 2027520, 3855986196480, 8359491805553413324800, 79457890145647634305213865656320000, 12897878211365028383150895090566532213003150950400000, 140613650417826346093374124598539442743630963394643403845144815232614400000

Offset: 1

Clark Kimberling, Jan 02 2012

nonn

Each term divides its successor, as in A203480.

G. C. Greubel, Table of n, a(n) for n = 1..21

Cf. A000225, A093883, A203305, A203480, A203481.

[(&*[(&*[2^j+2^k-2: k in [1..j]])/(2^(j+1)-2): j in [1..n]]): n in [1..15]]; // G. C. Greubel, Aug 28 2023

a:= n-> mul(mul(2^i+2^j-2, i=1..j-1), j=2..n):
seq(a(n), n=1..12);  # Alois P. Heinz, Jul 23 2017

(* First program *)
f[j_]:= 2^j -1; z = 15;
v[n_]:= Product[Product[f[k] + f[j], {j,k-1}], {k,2,n}]
Table[v[n], {n,z}]               (* A203479 *)
Table[v[n+1]/v[n], {n,z-1}]      (* A203480 *)
Table[v[n+1]/(4*v[n]), {n,z-1}]  (* A203481 *)
(* Second program *)
Table[Product[2^j +2^k -2, {j,n}, {k,j-1}], {n,15}] (* G. C. Greubel, Aug 28 2023 *)

[product(product(2^j+2^k-2 for k in range(1,j)) for j in range(1,n+1)) for n in range(1,16)] # G. C. Greubel, Aug 28 2023

a(n) ~ c * 2^((n-1)*n*(n+1)/3) / QPochhammer(1/2, 1/4)^(n-1), where c = 0.0732262905669624786298393270254722268761908164083517721484477901776137... - Vaclav Kotesovec, Aug 09 2025

Name edited by Alois P. Heinz, Jul 23 2017

A203477 a(n) = Product_{0 <= i < j <= n-1} (2^i + 2^j).

Original entry on oeis.org

1, 3, 90, 97200, 14276736000, 1107198567383040000, 178601637561927097909248000000, 237856509917156074017606774172522905600000000, 10420480393274493153643458442091600404477248333907230720000000000

Offset: 1

Clark Kimberling, Jan 02 2012

Each term divides its successor, as in A203478.

G. C. Greubel, Table of n, a(n) for n = 1..21

Cf. A000079, A093883, A164051, A203305, A203478.

[(&*[(&*[2^j + 2^k: k in [0..j]])/2^(j+1): j in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Aug 28 2023

a:= n-> mul(mul(2^i+2^j, i=0..j-1), j=1..n-1):
seq(a(n), n=1..10);  # Alois P. Heinz, Jul 23 2017

(* First program *)
f[j_]:= 2^(j-1); z = 13;
v[n_]:= Product[Product[f[k] + f[j], {j,k-1}], {k,2,n}]
Table[v[n], {n,z}]                       (* A203477 *)
Table[v[n+1]/v[n], {n,z-1}]              (* A203478 *)
Table[v[n]*v[n+2]/(2*v[n+1]^2), {n,22}]  (* A164051 *)
(* Second program *)
Table[Product[(2^j^2)*QPochhammer[-1/2^j,2,j], {j,0,n-1}], {n,20}] (* G. C. Greubel, Aug 28 2023 *)

a(n)=prod(i=0,n-2,prod(j=i+1,n-1,2^i+2^j)) \\ Charles R Greathouse IV, Feb 16 2021

[product(product(2^j + 2^k for k in range(j)) for j in range(n)) for n in range(1,21)] # G. C. Greubel, Aug 28 2023

Name edited by Alois P. Heinz, Jul 23 2017

A335011 Decimal expansion of Product_{k>=1} (1 - 1/2^k)^k.

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, May 19 2020

			0.0996797312628799854646653560468473049634446150973175301264091855545606305...

Cf. A005329, A048651, A203305, A333938, A335010.

evalf(exp(-Sum(2^j / (j * (2^j - 1)^2), j=1..infinity)), 120);

PARI
```
prodinf(k=1, (1 - 1/2^k)^k)
```

Equals exp(-Sum_{j>=1} 2^j / (j * (2^j - 1)^2)).

Showing 1-7 of 7 results.

cp's OEIS Frontend

A203307 a(n) = v(n+1)/(2*v(n)), where v = A203305.

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A203465 a(n) = A203305(n)/A000178(n) where A000178 are superfactorials.

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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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A028365 Order of general affine group over GF(2), AGL(n,2).

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A203479 a(n) = Product_{1 <= i < j <= n} (2^i + 2^j - 2).

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A203477 a(n) = Product_{0 <= i < j <= n-1} (2^i + 2^j).

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