A203483 -id:A203483 - cp's OEIS frontend

A203482 a(n) = Product_{1 <= i < j <= n} (i! + j!).

1, 3, 168, 3276000, 877449500928000, 207244701437748852512194560000, 4000516840149319128119305958853265913416777728000000, 796608816253064941944831363792070377592412324940256242675178274726476775424000000000

Offset: 1

Clark Kimberling, Jan 03 2012

nonn

Each term divides its successor, as in A203483.

See A093883 for a guide to related sequences.

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

Cf. A000142, A203483, A203510, A306729, A325052.

[(&*[(&*[Factorial(j) + Factorial(k): k in [1..j]])/(2*Factorial(j)): j in [1..n]]): n in [1..20]]; // G. C. Greubel, Aug 29 2023

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

(* 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}]     (* A203483 *)
Table[v[n]/d[n], {n, 10}]        (* A203510 *)
(* Second program *)
Table[Product[j!+k!, {j,n}, {k,j-1}], {n,15}] (* G. C. Greubel, Aug 29 2023 *)

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

a(n) ~ c * n^(n^3/3 + n^2/4 - 7*n/12 + 5/8) * (2*Pi)^(n*(n-1)/4) / exp(4*n^3/9 - n^2/8 - n), where c = 0.488888619502150098591650327163991582267254151817880403495924251381414248582... - Vaclav Kotesovec, Nov 20 2023

Name edited by Alois P. Heinz, Jul 23 2017

A323717 a(n) = Product_{k=0..n} (n! + k!).

Original entry on oeis.org

2, 4, 36, 4704, 23400000, 7778123781120, 245221791787632844800, 980866487456532919096049664000, 647456833933936977045736601678008811520000, 89423837415458106416291101560480526982914768896000000000

Offset: 0

Vaclav Kotesovec, Jan 25 2019

nonn

G. C. Greubel, Table of n, a(n) for n = 0..29

Cf. A036740, A203308, A203483, A323702.

F:= Factorial; [(&*[F(n) + F(j): j in [0..n]]): n in [0..20]]; // G. C. Greubel, Aug 30 2023

Table[Product[n!+k!, {k, 0, n}], {n, 0, 10}]

f=factorial; [product(f(n) + f(k) for k in range(n+1)) for n in range(21)] # G. C. Greubel, Aug 30 2023

a(n) ~ 2^((n+3)/2) * Pi^((n+1)/2) * n^((n+1)*(2*n+1)/2) / exp(n^2 + n - 1/12).

a(n) ~ 2 * (n!)^(n+1). - Vaclav Kotesovec, Mar 28 2019

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

cp's OEIS Frontend

A203482 a(n) = Product_{1 <= i < j <= n} (i! + j!).

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A323717 a(n) = Product_{k=0..n} (n! + k!).

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

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Magma

Mathematica

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