A323575 -id:A323575 - cp's OEIS frontend

A303641 Decimal expansion of a constant related to the asymptotics of A323575.

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

Offset: 1

Vaclav Kotesovec, Jan 19 2019

			2.4736552566321294876378936942724280363620971232545793827877...

Cf. A323575.

Equals limit_{n->infinity} A323575(n) / (n^(n*(n+1)/2) * 2^(n^2/2) / exp(n^2/2)).

A323588 a(n) = Product_{k=1..n} (k^n + (n-k)^n).

Original entry on oeis.org

1, 1, 8, 2187, 55083008, 248292236328125, 287440081598682287308800, 136294854579772162759923622710449623, 32534104705262209051040075603284216686012438413312, 5686543339012978225006873713961872387810223003912610672810622880089

Offset: 0

Vaclav Kotesovec, Jan 18 2019

nonn

G. C. Greubel, Table of n, a(n) for n = 0..27

Cf. A323540-A323546, A323575, A323589, A323751.

[1] cat [(&*[k^n +(n-k)^n: k in [1..n]]): n in [1..10]]; // G. C. Greubel, Feb 08 2019

Table[Product[k^n+(n-k)^n, {k, 1, n}], {n, 0, 10}]

vector(10, n, n--; prod(k=1,n, k^n+(n-k)^n)) \\ G. C. Greubel, Feb 08 2019

[product(k^n +(n-k)^n for k in (1..n)) for n in (0..10)] # G. C. Greubel, Feb 08 2019

a(n) ~ c * 2^(n^2) * n^(n^2) / exp(n^2), where
c = 1.7567468186007109703792640049745420817202851050652253469714... if n is even,
c = 1.8080216158688347442204158454365469233524049331246880759722... if n is odd.

A323589 a(n) = Product_{k=1..n-1} (k^k + (n-k)^(n-k)).

Original entry on oeis.org

1, 1, 2, 25, 6272, 63473089, 35671256150400, 1706937496190389809801, 7511133178157708431911079116800, 4755809816953036991699151550498501702425129, 394143276257895110158515904775794405720952934400000000000

Offset: 0

Vaclav Kotesovec, Jan 18 2019

nonn

G. C. Greubel, Table of n, a(n) for n = 0..31

Cf. A323575, A323588.

[1,1] cat [(&*[k^k + (n-k)^(n-k): k in [1..n-1]]): n in [2..12]]; // G. C. Greubel, Feb 08 2019

Table[Product[k^k+(n-k)^(n-k), {k, 1, n-1}], {n, 0, 12}]

vector(12, n, n--; prod(k=1,n-1, k^k+(n-k)^(n-k))) \\ G. C. Greubel, Feb 08 2019

[product(k^k + (n-k)^(n-k) for k in (1..n-1)) for n in (0..12)] # G. C. Greubel, Feb 08 2019

a(n) ~ n^(3*n^2/4 - n) * 2^(n^2/4 + 7/6) / exp(3*n^2/8) if n is even.

a(n) ~ n^(3*n^2/4 - n + 1/4) * 2^(n^2/4 - 1/12) / exp(3*n^2/8 - 1/4) if n is odd.

A323702 a(n) = Product_{k=0..n} (k! + (n-k)!).

Original entry on oeis.org

2, 4, 18, 441, 122500, 585640000, 61740367761072, 176956326932345427600, 16411667387809544192807523072, 59483286633748316026134239331720597504, 9536532654533775992805729638288082189179486453760, 81298938207133741609860679855100783339855352530145447380582400

Offset: 0

Vaclav Kotesovec, Jan 24 2019

nonn

G. C. Greubel, Table of n, a(n) for n = 0..35

Cf. A000178, A323575.

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

Table[Product[k! + (n-k)!, {k, 0, n}], {n, 0, 12}]

a(n) = prod(k=0, n, k! + (n-k)!); \\ Michel Marcus, Jan 24 2019

[product(factorial(j)+factorial(n-j) for j in range(n+1)) for n in range(21)] # G. C. Greubel, Aug 30 2023

a(n) ~ 2^(n^2/4 + n + 11/6) * n^(3*n*(n+2)/4 + 1/2) * Pi^((n+1)/2) / exp(3*n*(3*n+4)/8) if n is even.

a(n) ~ 2^(n^2/4 + n + 7/12) * n^(3*(n+1)^2 / 4) * Pi^((n+1)/2) / exp(3*n*(3*n+4)/8) if n is odd.

A323751 a(n) = Product_{k=0..n} (k^n + (n-k)^n).

Original entry on oeis.org

2, 1, 32, 59049, 14101250048, 775913238525390625, 13410804447068120796679372800, 112244673425189306235795780017831813874289, 545831702006800417886454373052629612732034857946832699392

Offset: 0

Seiichi Manyama, Jan 26 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..27

Cf. A323540, A323541, A323542, A323543, A323544, A323545, A323546.

Cf. A323575, A323588.

[(&*[k^n +(n-k)^n: k in [0..n]]): n in [0..10]]; // G. C. Greubel, Feb 08 2019

Table[Product[k^n+(n-k)^n, {k,0,n}], {n,0,10}] (* G. C. Greubel, Feb 08 2019 *)

PARI
```
{a(n) = prod(k=0, n, k^n+(n-k)^n)}
```

[product(k^n +(n-k)^n for k in (0..n)) for n in (0..10)] # G. C. Greubel, Feb 08 2019

a(n) = n^n * A323588(n). - Vaclav Kotesovec, Feb 08 2019

A366305 a(n) = Product_{k=1..n} (k^n + (k-1)^n).

Original entry on oeis.org

1, 5, 315, 555713, 47705305725, 305469864195354625, 207095306530955763265880535, 20017329298655447986400838721630926977, 357361761140807273279996172600335233468472149678425, 1481824279740988988264353294673429995981921700740921435758587890625

Offset: 1

Vaclav Kotesovec, Oct 06 2023

nonn

Cf. A036740, A323575, A323588, A323589, A366306.

Table[Product[k^n + (k-1)^n, {k, 1, n}], {n, 1, 10}]

a(n) = (n!)^n * Product_{k=1..n} (1 + (1 - 1/k)^n).
a(n) ~ n!^n * d^n, where d = exp(Integral_{x=0..1} log(1 + exp(-1/x)) dx) = 1.14183186235785012136459060138978468902610644657603999829892450823456733...
a(n) ~ (2*Pi)^(n/2) * d^n * n^(n*(2*n+1)/2) / exp(n^2 - 1/12).

A366271 Decimal expansion of limit_{n->oo} Product_{k=1..n} ((k/n)^(k/n) + (1 - k/n)^(k/n))^(1/n).

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Oct 06 2023

Limit_{n->oo} Product_{k=1..n} (k/n)^(k/n^2) = exp(-1/4).

			1.38489201265986890417861106075712813583048148929763977709475...

Cf. A323575.

RealDigits[Exp@NIntegrate[Log[1+(1/r-1)^r], {r, 0, 1}, WorkingPrecision->120] * Exp[-1/4], 10, 105][[1]]

Equals exp(-1/4 + Integral_{x=0..1} log(1 + (1/x - 1)^x) dx).

Conjecture: Limit_{n->oo} (1/A366271^n) * Product_{k=1..n} ((k/n)^(k/n) + (1 - k/n)^(k/n)) = 1/sqrt(2).

A366306 a(n) = Product_{k=1..n} (k^n - (k-1)^n).

Original entry on oeis.org

1, 3, 133, 170625, 10733002621, 50465283999665535, 25145494699347449245677097, 1787473773567267792523164108726890625, 23480751910878672340765325385856840967957995534681, 71672834655019406921956925590632596034005848922160549420728589375

Offset: 1

Vaclav Kotesovec, Oct 06 2023

nonn

Cf. A036740, A323575, A323588, A323589, A366305.

Table[Product[k^n - (k-1)^n, {k, 1, n}], {n, 1, 10}]

a(n) = (n!)^n * Product_{k=1..n} (1 - (1 - 1/k)^n).
a(n) ~ n!^n * d^n, where d = exp(Integral_{x=0..1} log(1 - exp(-1/x)) dx) = 0.84207793096051704199642805288991601369639823969574423397520945175552718...
a(n) ~ (2*Pi)^(n/2) * d^n * n^(n*(2*n+1)/2) / exp(n^2 - 1/12).

cp's OEIS Frontend

A303641 Decimal expansion of a constant related to the asymptotics of A323575.

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A323588 a(n) = Product_{k=1..n} (k^n + (n-k)^n).

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A323589 a(n) = Product_{k=1..n-1} (k^k + (n-k)^(n-k)).

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A323702 a(n) = Product_{k=0..n} (k! + (n-k)!).

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A323751 a(n) = Product_{k=0..n} (k^n + (n-k)^n).

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A366305 a(n) = Product_{k=1..n} (k^n + (k-1)^n).

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A366271 Decimal expansion of limit_{n->oo} Product_{k=1..n} ((k/n)^(k/n) + (1 - k/n)^(k/n))^(1/n).

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A366306 a(n) = Product_{k=1..n} (k^n - (k-1)^n).

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