A093883 -id:A093883 - cp's OEIS frontend

A203483 a(n) = v(n+1)/v(n), where v = A203482.

3, 56, 19500, 267841728, 236189890379520, 19303349192505048268800, 199126474924007956512865886208000, 339543987407937097660189431863908761600000000, 121553118121801544803671246298148699436481551316864204800000

Offset: 1

Clark Kimberling, Jan 03 2012

nonn

G. C. Greubel, Table of n, a(n) for n = 1..29

Cf. A000142, A093883, A203482, A203510, A323717.

[(&*[Factorial(k) + Factorial(n+1): k in [1..n]]): n in [1..16]]; // G. C. Greubel, Aug 29 2023

(* First program *)
f[j_]:= j!; z = 10;
v[n_]:= Product[Product[f[k] + f[j], {j,k-1}], {k,2,n}]
d[n_]:= Product[(i-1)!, {i, n}]   (* A000178 *)
Table[v[n], {n,z}]                (* A203482 *)
Table[v[n+1]/v[n], {n,z-1}]       (* this sequence *)
Table[v[n]/d[n], {n,10}]          (* A203510 *)
(* Second program *)
Table[Product[k!+(n+1)!, {k,n}], {n,15}] (* G. C. Greubel, Aug 29 2023 *)

[product(factorial(k) + factorial(n+1) for k in range(1,n+1)) for n in range(1,16)] # G. C. Greubel, Aug 29 2023

a(n) = Product_{k=1..n} (k! + (n+1)!). - G. C. Greubel, Aug 29 2023
From Vaclav Kotesovec, Nov 20 2023: (Start)
a(n) ~ (n+1)!^n.
a(n) ~ (2*Pi)^(n/2) * n^(n^2 + 3*n/2) / exp(n^2 - 13/12). (End)

A203511 a(n) = Product_{1 <= i < j <= n} (t(i) + t(j)); t = A000217 = triangular numbers.

Original entry on oeis.org

1, 1, 4, 252, 576576, 87178291200, 1386980110791475200, 3394352757964564324299571200, 1760578659300452732262852600316664217600, 255323290537547288382098619855584488593426606981120000

Offset: 0

Clark Kimberling, Jan 03 2012

nonn

Each term divides its successor, as in A203512.

See A093883 for a guide to related sequences.

Cf. A000217, A203512, A293290, A324403, A324443.

t:= n-> n*(n+1)/2:
a:= n-> mul(mul(t(i)+t(j), i=1..j-1), j=2..n):
seq(a(n), n=0..12);  # Alois P. Heinz, Jul 23 2017

f[j_] := j (j + 1)/2; z = 15;
v[n_] := Product[Product[f[k] + f[j], {j, 1, k - 1}], {k, 2, n}]
Table[v[n], {n, 1, z}]               (* A203511 *)
Table[v[n + 1]/v[n], {n, 1, z - 1}]  (* A203512 *)
Table[Product[k*(k+1)/2 + j*(j+1)/2, {k, 1, n}, {j, 1, k-1}], {n, 0, 10}] (* Vaclav Kotesovec, Sep 07 2023 *)

a(n) ~ c * 2^n * exp(n^2*(Pi/4 - 3/2) + n*(Pi/2 + 1)) * n^(n^2 - n - 2 - Pi/8), where c = 0.2807609661547466473998991675307759198889389396430915721129636653... - Vaclav Kotesovec, Sep 07 2023

Name edited by Alois P. Heinz, Jul 23 2017

a(0)=1 prepended by Alois P. Heinz, Jul 29 2017

A203516 a(n) = Product_{1 <= i < j <= n} 2*(i+j-1).

Original entry on oeis.org

1, 4, 192, 184320, 4954521600, 4794391461888000, 204135216112950312960000, 451965950843675288237663846400000, 60040562704967329457107799785403842560000000, 542366306792798635131534558788357929673196306432000000000

Offset: 1

Clark Kimberling, Jan 03 2012

nonn

Each term divides its successor, as in A034910.

See A093883 for a guide to related sequences.

G. C. Greubel, Table of n, a(n) for n = 1..33

Cf. A000178, A005408, A034910, A093883, A203516, A203517.

[2^Binomial(n,2)*(&*[Factorial(2*k)/Factorial(k): k in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Feb 19 2024

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

f[j_] := 2 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}]                   (* A203516 *)
Table[v[n + 1]/(4 v[n]), {n, 1, z - 1}]  (* A034910 *)
Table[v[n]/d[n], {n, 1, 20}]             (* A203517 *)
Table[2^(-1/24 - 3*n/2 + 3*n^2/2) * Glaisher^(3/2) * Pi^(1/4 - n/2) * BarnesG[1/2 + n]/E^(1/8), {n, 1, 10}] (* Vaclav Kotesovec, Sep 01 2023 *)

a(n) = my(pd=1); for(j=1, n, for(i=1, j-1, pd=pd*2*(i+j-1))); pd \\ Felix Fröhlich, Jul 23 2017

[2^binomial(n,2)*product(factorial(2*k)/factorial(k) for k in range(n)) for n in range(1,21)] # G. C. Greubel, Feb 19 2024

a(n) ~ sqrt(A) * 2^(-7/24 - n + 3*n^2/2) * exp(-1/24 + n/2 - 3*n^2/4) * n^(1/24 - n/2 + n^2/2), where A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Sep 01 2023
From G. C. Greubel, Feb 19 2024: (Start)
a(n) = BarnesG(n+1)*A203517(n).
a(n) = 2^binomial(n,2) * Product_{j=1..n-1} (2j)!/j!. (End)

Name edited by Alois P. Heinz, Jul 23 2017

A203527 a(n) = Product_{1 <= i < j <= n} (A018252(i) + A018252(j)); A018252 = nonprime numbers.

Original entry on oeis.org

1, 5, 350, 529200, 17542980000, 14783258730240000, 511420331138811494400000, 871980665589501641034301440000000, 60150685659205753788492548338089984000000000, 182771197941564481989784945231570147139911680000000000000

Offset: 1

Clark Kimberling, Jan 03 2012

nonn

Each term divides its successor, as in A203528. It is conjectured that each term is divisible by the corresponding superfactorial, A000178(n); as in A203529. See A093883 for a guide to related sequences.

Cf. A018252, A203415, A203528, A203529.

b:= proc(n) option remember; local k; if n=1 then 1
      else for k from 1+b(n-1) while isprime(k) do od; k fi
    end:
a:= n-> mul(mul(b(i)+b(j), i=1..j-1), j=2..n):
seq(a(n), n=1..10);  # Alois P. Heinz, Jul 23 2017

t = Table[If[PrimeQ[k], 0, k], {k, 1, 100}];
nonprime = Rest[Union[t]]              (* A018252 *)
f[j_] := nonprime[[j]]; z = 20;
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}]                 (* A203527 *)
Table[v[n + 1]/v[n], {n, 1, z - 1}]    (* A203528 *)
Table[v[n]/d[n], {n, 1, 20}]           (* A203529 *)

Name edited by Alois P. Heinz, Jul 23 2017

A203530 a(n) = Product_{1 <= i < j <= n} (c(i) + c(j)); c = A002808 = composite numbers.

Original entry on oeis.org

1, 10, 1680, 5569200, 426645273600, 1135354270482432000, 129053267560513803386880000, 556394398742051964595520667648000000, 99449133623220179596974346585642106880000000000

Offset: 1

Clark Kimberling, Jan 03 2012

nonn

Each term divides its successor, as in A203530.
It is conjectured that each term is divisible by the corresponding superfactorial, A000178(n); as in A203533.
See A093883 for a guide to related sequences.

Cf. A002808, A203418, A203532, A203533.

c:= proc(n) option remember; local k; if n=1 then 4
      else for k from 1+c(n-1) while isprime(k) do od; k fi
    end:
a:= n-> mul(mul(c(i)+c(j), i=1..j-1), j=2..n):
seq(a(n), n=1..10);  # Alois P. Heinz, Jul 23 2017

t = Table[If[PrimeQ[k], 0, k], {k, 1, 100}];
composite = Rest[Rest[Union[t]]]       (* A002808 *)
f[j_] := composite[[j]]; z = 20;
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}]                 (* A203530 *)
Table[v[n + 1]/v[n], {n, 1, z - 1}]    (* A203532 *)
Table[v[n]/d[n], {n, 1, 20}]           (* A203533 *)

Name edited by Alois P. Heinz, Jul 23 2017

A203675 Vandermonde sequence using x^2 - xy + y^2 applied to (1,4,9,...,n^2).

Original entry on oeis.org

1, 13, 57889, 560058939856, 42130404012097952586256, 65111467563626175389271488157658681344, 4528499444374253250530486688998183592108605307719698157568

Offset: 1

Clark Kimberling, Jan 04 2012

nonn

See A093883 for a discussion and guide to related sequences.

Cf. A203673, A367668.

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}]          (* A203675 *)
Table[v[n + 1]/v[n], {n, 1, z}] (* A203676 *)

a(n) ~ c * (2 + sqrt(3))^(sqrt(3)*n*(n+1)/2) * n^(2*n^2 - 2*n - 3/2) / exp(3*n^2 - Pi*n*(n+1)/4 - 2*n), where c = 0.07463795295314976973866568785704370572893158254239607676544741150586459722... - Vaclav Kotesovec, Nov 25 2023

A203677 Vandermonde sequence using x^2 + y^2 applied to (1,4,9,...,n^2).

Original entry on oeis.org

1, 17, 135218, 3185418047264, 795022479172023183220864, 5554004683279652358469137440150614769664, 2378852972988348412358457063032448409092378064835941488918528

Offset: 1

Clark Kimberling, Jan 04 2012

nonn

See A093883 for a discussion and guide to related sequences.

Cf. A324437.

f[j_] := j^2; z = 12;
u[n_] := Product[f[j]^2 + f[k]^2, {j, 1, k - 1}]
v[n_] := Product[u[n], {k, 2, n}]
Table[v[n], {n, 1, z}]          (* A203677 *)
Table[v[n + 1]/v[n], {n, 1, z}] (* A203678 *)

a(n) ~ c * 2^(n^2/2 - 1) * (1 + sqrt(2))^(n*(n+1)/sqrt(2)) * exp((Pi/2^(3/2) - 3)*n^2 + (Pi/2^(3/2) + 2)*n) * n^(2*n^2 - 2*n - 3/2), where c = 0.154147406559582639039828423669556073435424655001221440918550218582474208... - Vaclav Kotesovec, Sep 08 2023

A203685 Vandermonde sequence using x^2 + xy + y^2 applied to (1,2,6,...,n!).

Original entry on oeis.org

1, 7, 15652, 4466072310336, 257010442859540072735404130304, 12039114411811108434329752218564963041089979745446343475200

Offset: 1

Clark Kimberling, Jan 04 2012

nonn

See A093883 for a discussion and guide to related sequences.

f[j_] := j!; z = 8;
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}]          (* A203685 *)
Table[v[n + 1]/v[n], {n, 1, z}] (* A203686 *)

A203689 Vandermonde sequence using x^2 + y^2 applied to (1,2,6,...,n!).

Original entry on oeis.org

1, 5, 7400, 1515608208000, 67968144879431450715389952000, 2618461740951506934762583974466059060153960058218086400000

Offset: 1

Clark Kimberling, Jan 04 2012

nonn

See A093883 for a discussion and guide to related sequences.

f[j_] := j!; z = 8;
u[n_] := Product[f[j]^2 + f[k]^2, {j, 1, k - 1}]
v[n_] := Product[u[n], {k, 2, n}]
Table[v[n], {n, 1, z}]          (* A203689 *)
Table[v[n + 1]/v[n], {n, 1, z}] (* A203690 *)

a(n) ~ c * n^(2*n^3/3 + n^2/2 - 7*n/6 - 3/4) * (2*Pi)^(n*(n-1)/2) / exp(8*n^3/9 - n^2/4 - 2*n), where c = 0.574053995428296820717722993471497472923951255369819486673614046900594637219... - Vaclav Kotesovec, Nov 21 2023

A203752 Vandermonde sequence using x^2 - xy + y^2 applied to (0,1,1,2,2,...,floor(n/2)).

Original entry on oeis.org

1, 1, 1, 36, 5184, 112021056, 21785966991936, 1433615623503400157184, 1509414758014670876688343105536, 956401356293432867934306416285626820198400, 15149970368698147242050701825966625432586471604224000000

Offset: 1

Clark Kimberling, Jan 05 2012

nonn

See A093883 for a discussion and guide to related sequences.

f[j_] := Floor[j/2]; z = 20;
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}]          (* A203752 *)
Table[v[n + 1]/v[n], {n, 1, z}] (* A203753 *)
Table[Sqrt[v[n + 1]/v[n]], {n, 1, z}]  (* A203754 *)

cp's OEIS Frontend

A203483 a(n) = v(n+1)/v(n), where v = A203482.

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A203511 a(n) = Product_{1 <= i < j <= n} (t(i) + t(j)); t = A000217 = triangular numbers.

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A203516 a(n) = Product_{1 <= i < j <= n} 2*(i+j-1).

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A203527 a(n) = Product_{1 <= i < j <= n} (A018252(i) + A018252(j)); A018252 = nonprime numbers.

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A203530 a(n) = Product_{1 <= i < j <= n} (c(i) + c(j)); c = A002808 = composite numbers.

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A203675 Vandermonde sequence using x^2 - xy + y^2 applied to (1,4,9,...,n^2).

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A203677 Vandermonde sequence using x^2 + y^2 applied to (1,4,9,...,n^2).

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A203685 Vandermonde sequence using x^2 + xy + y^2 applied to (1,2,6,...,n!).

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