A093883 -id:A093883 - cp's OEIS frontend

A203763 Vandermonde sequence using x^2 - xy + y^2 applied to (1,1,2,2,...,[n/2]).

1, 1, 9, 324, 777924, 16810159716, 69136555917409344, 4549499535875623543259136, 115306876482136485813839025883201536, 73061199694724861313901918528002630365482598400

Offset: 1

Clark Kimberling, Jan 05 2012

nonn

See A093883 for a discussion and guide to related sequences.

f[j_] := Floor[(j + 1)/2]; z = 16;
u := Product[f[j]^2 - f[j] f[k] + f[k]^2, {j, 1, k - 1}]
v[n_] := Product[u, {k, 2, n}]
Table[v[n], {n, 1, z}]        (* A203763 *)
Table[v[n + 1]/v[n], {n, 1, z}]  (* A203764 *)
Table[Sqrt[v[n + 1]/v[n]], {n, 1, 20}]  (* A203765 *)

A203474 a(n) = A203472(n) / A000178(n-1), where A000178 are the superfactorials.

Original entry on oeis.org

1, 7, 252, 41580, 29729700, 89278289100, 1104908105901600, 55674109640169820800, 11329124570678156834592000, 9258047307912482983660236480000, 30262334718212007877669234596364800000

Offset: 1

Clark Kimberling, Jan 02 2012

nonn

G. C. Greubel, Table of n, a(n) for n = 1..55

Cf. A000178, A093883, A203472, A203473.

[(&*[ Binomial(2*j+3, j+4): j in [1..n]]): n in [1..20]]; // G. C. Greubel, Aug 27 2023

(* First program *)
f[j_]:= j+2; z=16;
v[n_]:= Product[Product[f[k] + f[j], {j,k-1}], {k,2,n}];
d[n_]:= Product[(i-1)!, {i,n}]  (* A000178(n-1) *)
Table[v[n], {n,z}]              (* A203472 *)
Table[v[n+1]/v[n], {n,z-1}]     (* A203473 *)
Table[v[n]/d[n], {n,20}]        (* A203474 *)
(* Second program *)
Table[Product[Binomial[2*j+3, j+4], {j,n}], {n,20}] (* G. C. Greubel, Aug 27 2023 *)

[product( binomial(2*j+5,j+5) for j in range(n) ) for n in range(1,20)] # G. C. Greubel, Aug 27 2023

a(n) ~ 3*A^(3/2) * 2^(n^2 + 4*n + 185/24) * exp(n/2 - 1/8) / (Pi^(n/2 + 3/2) * n^(n/2 + 59/8)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 09 2021
From G. C. Greubel, Aug 27 2023: (Start)
a(n) = Product_{j=1..n} binomial(2*j+3, j+4).
a(n) = (18*2^(n+2)^2/Pi^(n/2))*BarnesG(n+3)*BarnesG(n+7/2)/( BarnesG(n +1)*BarnesG(n+6)*BarnesG(7/2)). (End)

Definition corrected by Vaclav Kotesovec, Apr 09 2021

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

A203480 a(n) = v(n+1)/v(n), where v = A203479.

Original entry on oeis.org

4, 80, 6336, 1901824, 2167925760, 9505110118400, 162323441859870720, 10902076148767162433536, 2898720791385603198124032000, 3064112360434477703904869089280000, 12909951234577776926559241120412860416000

Offset: 1

Clark Kimberling, Jan 02 2012

nonn

G. C. Greubel, Table of n, a(n) for n = 1..55

Cf. A093883, A203307, A203479.

Cf. A079555.

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

(* 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^(n+1) +2^k -2, {k,n}], {n,20}] (* G. C. Greubel, Aug 28 2023 *)

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

a(n) = Product_{k=1..n} (2^k + 2^(n+1) - 2). - G. C. Greubel, Aug 28 2023

a(n) ~ c * 2^(n*(n+1)), where c = 1/QPochhammer(1/2, 1/4) = A079555 = 2.3842310290313717... - Vaclav Kotesovec, Aug 09 2025

A203481 a(n) = v(n+1)/(4*v(n)), where v = A203479.

Original entry on oeis.org

1, 20, 1584, 475456, 541981440, 2376277529600, 40580860464967680, 2725519037191790608384, 724680197846400799531008000, 766028090108619425976217272320000, 3227487808644444231639810280103215104000

Offset: 1

Clark Kimberling, Jan 02 2012

nonn

G. C. Greubel, Table of n, a(n) for n = 1..55

Cf. A203479, A203480, A093883.

Cf. A079555.

[(&*[2^j + 2^(n+1) - 2: j in [1..n]])/4: n in [1..20]]; // G. C. Greubel, Aug 28 2023

(* 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^(n+1) +2^k -2, {k,n}]/4, {n,20}] (* G. C. Greubel, Aug 28 2023 *)

[product(2^j+2^(n+1)-2 for j in range(1,n+1))/4 for n in range(1,21)] # G. C. Greubel, Aug 28 2023

a(n) = (1/4)*Product_{k=1..n} (2^k + 2^(n+1) - 2). - G. C. Greubel, Aug 28 2023

a(n) ~ c * 2^(n*(n+1)-2), where c = 1/QPochhammer(1/2, 1/4) = A079555 = 2.3842310290313717... - Vaclav Kotesovec, Aug 09 2025

A203510 a(n) = A203482(n) / A000178(n).

Original entry on oeis.org

1, 3, 84, 273000, 3046699656000, 5996663814749677445376000, 160771799453017261771769947549079938007040000, 6351968589735888467306807912855132014808202373395298410963148996608000000

Offset: 1

Clark Kimberling, Jan 03 2012

nonn

It is conjectured that every term of the sequence is an integer.

G. C. Greubel, Table of n, a(n) for n = 1..16

Cf. A000142, A000178, A074962, A093883, A203482, A203483.

BarnesG:= func< n | (&*[Factorial(k): k in [0..n-2]]) >;
A203510:= func< n | n eq 1 select 1 else (&*[(&*[Factorial(j) + Factorial(k): k in [1..j-1]]): j in [2..n]])/BarnesG(n+1) >;
[A203510(n): n in [1..13]]; // G. C. Greubel, Feb 24 2024

f[j_] := j!; z = 10;
v[n_] := Product[Product[f[k] + f[j], {j, 1, k - 1}], {k, 2, n}]
d[n_] := Product[(i - 1)!, {i, 1, n}]  (* A000178 *)
Table[v[n], {n, 1, z}]                 (* A203482 *)
Table[v[n + 1]/v[n], {n, 1, z - 1}]    (* A203483 *)
Table[v[n]/d[n], {n, 1, 10}]           (* this sequence *)
Table[Product[j! + k!, {j, 1, n}, {k, 1, j-1}] / BarnesG[n+1], {n, 1, 10}] (* Vaclav Kotesovec, Nov 20 2023 *)

def BarnesG(n): return product(factorial(j) for j in range(1,n-1))
def A203510(n): return product(product(factorial(j)+factorial(k) for k in range(1,j)) for j in range(1,n+1))/BarnesG(n+1)
[A203510(n) for n in range(1,14)] # G. C. Greubel, Feb 24 2024

a(n) ~ c * A * n^(n^3/3 - n^2/4 - 7*n/12 + 17/24) * (2*Pi)^(n^2/4 - 3*n/4) / exp(4*n^3/9 - 7*n^2/8 - n + 1/12), where A is the Glaisher-Kinkelin constant A074962 and c = 0.488888619502150098591650327163991582267254151817880403495924251381414248582... (from A203482). - Vaclav Kotesovec, Nov 20 2023

A203514 Vandermonde sequence using x^2 + xy + y^2 applied to (1,3,5,...,2n-1).

Original entry on oeis.org

1, 13, 19747, 9692360769, 3007399210929125649, 974293562642242789006882422237, 492054106680311213790793284550789224583937187, 540918035328864126384298694506545526610971072745364246025968225

Offset: 1

Clark Kimberling, Jan 04 2012

nonn

See A093883 for a discussion and guide to related sequences.

f[j_] := 2 j - 1; 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}]          (* A203514 *)
Table[v[n + 1]/v[n], {n, 1, z}] (* A203515 *)

A203519 a(n) = v(n+1)/v(n), where v=A203518.

Original entry on oeis.org

3, 20, 336, 12870, 1270080, 311323584, 197399802600, 321880885724160, 1365311591573529600, 15068868587132753685600, 434169705562891299584593920, 32678748925653999616045678080000, 6431834564578466234122576826339121600

Offset: 1

Clark Kimberling, Jan 03 2012

nonn

Cf. A203518, A000045, A093883.

f[j_] := Fibonacci[j + 1]; z = 15;
v[n_] := Product[Product[f[k] + f[j], {j, 1, k - 1}], {k, 2, n}]
d[n_] := Product[(i - 1)!, {i, 1, n}] (* A000178(n-1) *)
Table[v[n], {n, 1, z}]                (* A203518 *)
Table[v[n + 1]/v[n], {n, 1, z - 1}]   (* A203519 *)
Table[v[n]/d[n], {n, 1, 20}]          (* A203520 *)

a(n) ~ c * phi^(n*(n+2) + 5/6) / 5^(n/2 + 1/4), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio and c = 3.990771264633156107481636998828550132941483550455485713064916076346986459357... - Vaclav Kotesovec, Apr 09 2021

A203520 v(n)/A000178(n); v=A203518 and A000178=(superfactorials).

Original entry on oeis.org

1, 3, 30, 1680, 900900, 9535125600, 4122929827336320, 161481256755920962660800, 1289130207153926967849156327590400, 4850265693548396005370498087328884780717568000, 20141097979706537636828034511787661382412368790843921121216000

Offset: 1

Clark Kimberling, Jan 03 2012

nonn

It is conjectured that every term of A203520 is an integer.

Cf. A203518, A000045, A000178, A093883.

f[j_] := Fibonacci[j + 1]; z = 15;
v[n_] := Product[Product[f[k] + f[j], {j, 1, k - 1}], {k, 2, n}]
d[n_] := Product[(i - 1)!, {i, 1, n}] (* A000178 *)
Table[v[n], {n, 1, z}]                (* A203518 *)
Table[v[n + 1]/v[n], {n, 1, z - 1}]   (* A203519 *)
Table[v[n]/d[n], {n, 1, 20}]          (* A203520 *)

A203522 a(n) = v(n+1)/v(n), where v = A203521.

Original entry on oeis.org

1, 5, 56, 1080, 52416, 2073600, 168537600, 9963233280, 1122750720000, 234209649623040, 22056529195008000, 5634088861040640000, 1027073825689514803200, 132063784535221862400000, 27917533370401003929600000

Offset: 0

Clark Kimberling, Jan 03 2012

nonn

Robert Israel, Table of n, a(n) for n = 0..290
Robert Israel, Table of n, a(n) for n = 1..290

Cf. A203522, A203523, A000040, A093883.

f:= proc(n) local i; mul(ithprime(i)+ithprime(n+1),i=1..n) end proc:
map(f, [$1..30]); # Robert Israel, Jan 28 2025

f[j_] := Prime[j]; z = 15;
v[n_] := Product[Product[f[k] + f[j], {j, 1, k - 1}], {k, 2, n}]
d[n_] := Product[(i - 1)!, {i, 1, n}] (* A000178 *)
Table[v[n], {n, 1, z}]                (* A203521 *)
Table[v[n + 1]/v[n], {n, 1, z - 1}]   (* A203522 *)
Table[v[n]/d[n], {n, 1, 20}]          (* A203523 *)

a(n) = Product_{i=1..n} (prime(i)+prime(n+1)). - Robert Israel, Jan 28 2025

a(0) = 1 inserted by Robert Israel, Jan 29 2025

cp's OEIS Frontend

A203763 Vandermonde sequence using x^2 - xy + y^2 applied to (1,1,2,2,...,[n/2]).

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A203474 a(n) = A203472(n) / A000178(n-1), where A000178 are the superfactorials.

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

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A203480 a(n) = v(n+1)/v(n), where v = A203479.

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A203481 a(n) = v(n+1)/(4*v(n)), where v = A203479.

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A203510 a(n) = A203482(n) / A000178(n).

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Magma

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A203514 Vandermonde sequence using x^2 + xy + y^2 applied to (1,3,5,...,2n-1).

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Mathematica

A203519 a(n) = v(n+1)/v(n), where v=A203518.

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