A325052 -id:A325052 - cp's OEIS frontend

A306729 a(n) = Product_{i=0..n, j=0..n} (i! + j!).

2, 16, 5184, 9559130112, 109045776752640000000000, 27488263744928988967331390258832998400000000000, 1147897050240877062218236820013018349788772091106840426434074807527014400000000000000

Offset: 0

Vaclav Kotesovec, Mar 06 2019

nonn

Cf. A000178, A055462, A079478, A203482, A217757, A323717, A324403, A325052.

Table[Product[i! + j!, {i, 0, n}, {j, 0, n}], {n, 0, 7}]
Clear[a]; a[n_] := a[n] = If[n == 0, 2, a[n-1] * Product[k! + n!, {k, 0, n}]^2 / (2*n!)]; Table[a[n], {n, 0, 7}] (* Vaclav Kotesovec, Mar 27 2019 *)
Table[Product[Product[k! + j!, {k, 0, j}], {j, 1, n}]^2 / (2^(n-1) * BarnesG[n + 2]), {n, 0, 7}] (* Vaclav Kotesovec, Mar 27 2019 *)

from math import prod, factorial as f
def a(n): return prod(f(i)+f(j) for i in range(n) for j in range(n))
print([a(n) for n in range(1, 8)]) # Michael S. Branicky, Feb 16 2021

a(n) ~ c * 2^(n^2/2 + 2*n) * Pi^(n^2/2 + n) * n^(2*n^3/3 + 2*n^2 + 11*n/6 + 5/2) / exp(8*n^3/9 + 2*n^2 + n), where c = A324569 = 62.14398692334529025548974541735...

a(n) = a(n-1) * A323717(n)^2 / (2*n!). - Vaclav Kotesovec, Mar 28 2019

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

Original entry on oeis.org

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

A325053 a(n) = Product_{i=0..n, j=0..n} (i! + j! + 1).

Original entry on oeis.org

3, 81, 103680, 447180963840, 7014935716261432173527040, 1921470539412808834455592518302690305036517376000, 81601182941928855942156180258180656419177691149082352022004942698629910149621350400000

Offset: 0

Vaclav Kotesovec, Mar 26 2019

nonn

Cf. A107254, A306729, A325052.

Table[Product[i! + j! + 1, {i, 0, n}, {j, 0, n}], {n, 0, 7}]

a(n) = A306729(n) * Product_{i=0..n, j=0..n} (1 + 1/(i! + j!)).

a(n) ~ c * A324569 * 2^(n^2/2 + 2*n) * Pi^(n^2/2 + n) * n^(2*n^3/3 + 2*n^2 + 11*n/6 + 5/2) / exp(8*n^3/9 + 2*n^2 + n), where c = Product_{i>=0, j>=0} (1 + 1/(i! + j!)) = 71.32069635593350979104242285703294604508330622582076432053456223608...

cp's OEIS Frontend

A306729 a(n) = Product_{i=0..n, j=0..n} (i! + j!).

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

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A325053 a(n) = Product_{i=0..n, j=0..n} (i! + j! + 1).

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